CHAPTER VII. NUMERICAL CALCULATIONS. i6o. Engineering work leads to more or less extensive numerical calculations, when applying the general theoretical investigation to the specific cases which are under considera- tion. Of importance in such engineering calculation^ are : (a) The method of calculation. (5) The degree of exactness required in the calculation. (c) The intelligibility of the results. (d) The reliability of the calculation. a. Method of Calculation. Before beginning a more extensive calculation, it is desirable carefully to scrutinize and to investigate the method, to find the simplest way, since frequently by a suitable method and system of calculation the work can be reduced to a small frac- tion of what it would otherwise be, and what appear to be hopelessly complex calculations may thus be carried out quickly and expeditiously by a proper arrangement of the work. The most convenient way usually is the arrangement in tabular form. As example, consider the problem of calculating the regula- tion of a 6(),000-volt transmission line, of r = 60 ohms resist- ance, a;= 135 ohms inductive reactance, and 6 = 0.0012 conden- sive susceptance, for various values of non-inductive, inductive, and condensive load. Starting with the complete equations of the long-distance transmission line, as given in "Theory and Calculation of Transient Electric Phenomena and Oscillations," Section III, paragraph 9, and considering that for every one of the various power-factors, lag, and lead, a sufficient number of values 249 250 ENGINEERING MATHEMATICS. have to be calculated to give a curve, the amount of work appears hopelessly large. However, without loss of engineering exactness, the equa- tion of the transmission line can be simplified by approxima- tion, as discussed in Chapter V, paragraph 123, to the form. + ^/o 1+- ZY + F^oU+^^ (1) where Eo, h are voltage and current, respectively at the step- down end, El, I\ at the step-up end of the line; and Z = r—jx = Q^—\Zbj is the total line impedance; Y = g — jh= —0.0012/ is the total shunted line admittance. Herefrom follow the numerical values : ZY (60-135.f)(-0.0012i) ■^2 2 = 1 - 0.036./- 0.081 = 0.919 - 0.036/; ZY 1+- g- = 1 - 0.012/- 0.027 = 0.973 - 0.012/; ryi ZY Z 14--,- -»4'' = (60- 135/) (0.973 -0.012/) = 58.4-0.72/- 131.1/- 1.62 = 56.8-131.8/; = (-0.0012/) (0.973 -0.012/) = -0.001168/-0.0000144 = (-0.0144- 1.168/)10-2 hence, substituting in (1), the following equations may be written : J^i = (0.919-0.036/)J5;o+(56.8-131.8/)/o = A +5; 1 /i = (0.919 -0.036/)7o - (0.0144 +1.168/)^olO-3 = C-D. / (2> NUMERICAL CALCULATIONS, 251 i6i. Now the work of calculating a series of numerical values is continued in tabular form, as follows : 1. 100 PER CENT Power-factor. Eo=60 kv. at step-down end of line. A = (0.919-0.036/)£;o=55.1-2.2y kv. D = (0. 0 144+1. 168?) £?o 10- 3 = 0.9 + 70.1/ amp. Iq amp. Bkv. Ei = ei- -eJ2 = A+B. ei2 + g22 = g2. e — = tan e. ei 4-6. 0 0 55.1- - 2.2/ 3036+ 5 = 3041 55.1 -0.040 - 2.3 20 1.1- 2.6/ 56.2- - 4.8/ 3158+ 23 = 3181 56.4 -0.085 - 4.9 40 2.3- 5.3/ 57.4- - 9.5/ 3295+ 56 = 3351 57.9 -0.131 - 7.5 60 3.4- 7.9/ 58.5- -10.1/ 3422 + 102 = 3524 59.4 -0.173 - 9.9 80 4.5-10.5/ 59.6- -12.7/ 3552 + 161 = 3713 60.9 -0.213 -12.0 100 5.7-13.2/ 60.8- -15.4/ 3697 + 237 = 3934 62.7 -0.253 -14.2 120 6.8-15.8/ 61.9- -18.0/ 3832 + 324=4156 64.5 -0.291 -16.3 h amp. , C amp. Il = tl=/t2 = C-D tl2 + t22 = i2 i *i=tam A-i 2^t- i 4-e= 4-«'* Power- factor 0 0 -0.7-90.1/ 4914+1 = 4915 70.1 + 78 + 89.1 -90.9 -88.6 0.024 20 18.4-0.7/ 17.5-70.8/ 5013+ 306= 5319 72.9 -4.04 -76.3 -71.4 0.332 40 36.8-1.4/ 35.9-71.5/ 5112 + 1289= 6401 80.0 -1.99 -63.4 -55.9 0.558 60 55.1-2.2/ 54.2-72.3/ 5227 + 2938= 8165 90.4 -1.33 -53.1 -43.2 0.728 80 73.5-2.9/ 72.6-73.0/ 5329 + 5271=10600 103.0 -1.055 -45.2 -33.2 0.837 100 91.9-3.6/ 91.0-73.9/ 8281 + 5432=13713 117.1 -0.811 -39.1 -24.9 0.907 120 110.3-4.3/ 109.4-74.4/ 11969 + 5535=17504 132.3 -0.680 -34.1 -17.8 0.952 lead 61 = 60 kv. at step-up end of line. /o amp. Red. Factor, e 60 amp. kv. amp. Power-Factor. 0 0.918 0 65.5 76.4 0.024 20 0.940 21.3 63.8 77.5 0.332 40 0.965 41.4 62.1 82.9 0.558 60 0.990 60.6 60.6 91.4 0.728 80 1.015 78.8 59.1 101.5 0.837 100 1.045 95.7 57.5 112.3 0.907 120 1.075 111.7 55.8 122.8 0.952 lead Curves of Iq, e^, ij, cos d, plotted in Fig. 86. 252 ENGINEERING MATHEMATICS. 2. 90 Per Cent Power-Factor, Lag. cos ^ = 0.9; sin^ = \/l-0.92 = 0.436; /o = ^o(cos d+jsin ^)=2o(0.9 +0.436/); j&i = (0.919- 0.036j>o + (56.8- 131.8y) (0.9 +0.436j)^ = (0.919-0.036f)eo + (108.5-93.8j>'o = A+5'; 7i = (0.919-0.036f)(0.9+0.436y)io- (0.0144 + 1.168j>ol0-3 = (0.843 +0.366y)io- (0.0144 + 1.168j>olO-3 = C'-D, and now the table is calculated in the same manner as under 1. Then corresponding tables are calculated, in the same manner, for power-factor, =0.8 and =0.7, respectively, lag, and for power-factor =0.9, 0.8, 0.7, lead; that is, for cos ^+y sin 6' = 0.8+0.6y; 0.7+0.714/; 0.9-0.436/; 0.8-0.6/: 0.7-0.714/. Then curves are plotted for ail seven values of power-factor, from 0.7 lag to 0.7 lead. From these curves, for a number of values of io, for instance, to = 20, 40, 60, 80, 100, numerical values of ii, eo, cos d, are taken, and plotted as curves, which, for the same voltage ei = 60 at the step-up end, give ii, eo, and cos d, for the same value to, that is, give the regulation of the line at constant current output for varying power-factor. b. Accuracy of Calculation. 162. Not all engineering calculations require the same degree of accuracy. When calculating the efficiency of a large alternator it may be of importance to detcTmine whether it is 97.7 or 97.8 per cent, that is, an accuracy within one-tenth per cent may be required; in other cases, as for instance, when estimating the voltage which may be produced in an electric circuit by a line disturbance, it may be sufficient to NUMERICAL CALCULATIONS. 253 determine whether this voltage would be hmited to double the rxormal circuit voltage, or whether it might be 5 or 10 times the normal voltage. In general; according to the degree of accuracy, engineering calculations may be roughly divided into three classes : (a) Estimation of the magnitude of an effect; that is, determining approximate numerical values within 25, 50, or 100 per cent. Very frequently such very rough approximation is sufficient, and is all that can be expected or calculated. kv. ^^ ■-- Co . 65 ei ^^ y y '^^'^ > y y - ii^ ^ ^ ^ -"^ < / ^1 n^> / / J / / / / / / 2 3 4 3 6 b 8 3 1 X) 1^ a. e ■120 1001.00 80 0.-80 0.40 0.20 ^ 0 Fig. 86. Transmission Line Characteristics. For instance, when investigating the short-circuit current of an electric generating system, it is of importance to know whether this current is 3 or 4 times normal current, or whether it is 40 to 50 times normal current, but it is immaterial whether it is 45 to 46 or 50 times normal. In studying lightning phenomena, and, in general, abnormal voltages in electric systems, calculating the discharge capacity of lightning arres- ters, etc., the magnitude of the quantity is often sufficient. In 254 ^ ENGINEERING MATHEMATICS. calculating the critical speed of turbine alternators, or the natural period of oscillation of synchronous machines, the same applies, since it is of importance only to see that these speeds are sufficiently^ remote from the normal operating speed to give no trouble in operation. (b) Approximate calculation, requiring an accuracy of one or a few per cent only; a large part of engineering calcu- lations fall in this class, especially calculations in the realm of design. Although, frequently, a higher accuracy could be reached in the calculation proper, it would be of no value, since the data on which the calculations are based are sus- ceptible to variations beyond control, due to variation in the material, in the mechanical dimensions, etc. Thus, for instance, the exciting current of induction motors may vary by several per cent, due to variations of the length of air gap, so small as to be beyond the limits of constructive accuracy, and a calculation exact to a fraction of one per cent, while theoretically possible, thus would be practically useless, The calculation of the ampere-turns required for the shunt field excitation, or for the series field of a direct-current generator needs only moderate exactness, as variations in the magnetic material, in the speed regulation of the driving power, etc., produce differences amounting to several per cent. (c) Exact engineering calculations, as, for instance, the calculations of the efficiency of apparatus, the regulation of transformers, the characteristic curves of induction motors, etc. These are determined with an accuracy frequently amount- ing to one-tenth of one per cent and even greater. Even for most exact engineering calculations, the accuracy of the slide rule is usually sufficient, if intelligently used, that is, used so as to get the greatest accuracy. Thus, in dividing, for instance, 297 by 283 by the slide rule, the proper way is to divide 297-283 = 14 by 283, and to add the result to 1. This gives a greater accuracy than direct division. For accu- rate calculations, preferably the glass slide should not be used, but the result interpolated by the eye. 163. While the calculations are unsatisfactory, if not carried out with the degree of exactness which is feasible and desirable, it is equally wrong to give numerical values with a number of NUMERICAL CALCULATIONS. 255 ciphers greater than the method or the purpose of the calcula- tion warrants. For instance, if in the design of a direct-current generator, the calculated field ampere-turns are given as 9738, such a numerical value destroys the confidence in the work of the calculator or designer, as it implies an accuracy greater than possible, and thereby shows a lack of judgment. The number of ciphers in which the result of calculation is given should signify the exactness, In this respect two systems are in use: (a) Numerical values are given with one more decimal than warranted by the probable error of the result; that is, the decimal before the last is correct, but the last decimal may be wrong by several units. This method is usually employed in astronomy, physics, etc. (6) Numerical values are given with as many decimals as the accuracy of the calculation warrants; that is, the last decimal is probably correct within half a unit. For instance, an efficiency of 86 per cent means an efficiency between 85.5 and 86.5 per cent; an efficiency of 97.3 per cent means an efficiency between 97.25 and 97.35 per cent, etc. This system is generally used in engineering calculations. To get accuracy of the last decimal of the result, the calculations then must be carried out for one more decimal than given in the result. For instance, when calculating the efficiency by adding the various percentages of losses, data like the following may be given : Core loss 2.73 per cent i^r 1.06 '' Friction 0.93 " Total 4.72 Efficiency 100-4.72 = 95.38 Approximately 95.4 " It is obvious that throughout the same calculation the same degree of accuracy must be observed. It follows herefrom that the values : 2i; 2.5; 2.50; 2.500, 256 ENGINEERING MATHEMATICS. while mathematically equal, are not equal in their meaning as an engineering result : 2.5 means between 2.45 and 2.55; 2.50 means between 2.495 and 2.505; 2.500 means between 2.4995 and 2.5005; while 2i gives no clue to the accuracy of the value. Thus it is not permissible to add zeros, or drop, zeros at the end of numerical values, nor is it permissible, for instance, to replace fractions as 1/16 by 0.0625, without changing the meaning of the numerical value, as regards its accuracy. This is not always realized, and especially in the reduction of common fractions to decimals an unjustified laxness exists which impairs the reliability of the results. For instance, if in an arc lamp the arc length, for which the mechanism is adjusted, is stated to be 0.8125 inch, such a statement is ridiculous, as no arc lamp mechanism can control for one-tenth as great an accuracy as implied in this numerical value: the value is an unjustified translation from 13/16 inch. The principle thus should be adhered to, that all calcula- tions are carried out for one decimal more than the exactness required or feasible, and in the result the last decimal dropped; that is, the result given so that the last decimal is probably correct within half a unit. c. Intelligibility of Engineering Data. 164. In engineering calculations the value of the results mainly depends on the information derived from them, that is, on their intelligibility. To make the numerical results and their meaning as intelligible as possible, it thus is desirable, whenever a series of values are calculated, to carefully arrange them in tables and plot them in a curve or in cur^^s. The latter is necessary, since for most engineers the plotted curve gives a much better conception of the shape. and the variation of a quantity than numerical tables. Even where only a single point is required, as the core loss at full load, or the excitation of an electric generator at rated voltage, it is generally preferable to calculate a few NUMERICAL CALCULATIONS. 257 ■■ Volts — 100 0 2 0 4 0 6 0 8 1 0 Fig. 87. Compounding Curve. points near the desired value, so as to get at least a short piece of curve including the desired point. The main advantage, and foremost purpose of curve plotting thus is to show the shape of the function, and thereby give a clearer conception of it ; but for recording numerical values, and deriving numer- ical values from it, the plotted curve is inferior to the table, due to the limited accuracy possible in a plotted curve, and the further inaccuracy resulting when drawing a curve through the plotted cal- culated points. To some extent, the numerical values as taken from a plotted curve, depend on the particular kind of curve rule used in plotting the curve. In general, curves are used for two different purposes, and on the purpose for which the curve is plotted, should depend the method of plotting, as the scale, the zero values, etc. When curves are used to illustrate the shape of the function, so as to show how much and in what manner a quantity varies as function of another, large divisions of inconspicuous cross-section- ing are desirable, but it is -49o| essential that the cross- sectioning should extend to the zero values of the func- tion, even if the numerical values do not extend so far, since otherwise a wrong impression would be con- ferred. As illustrations are plotted in Figs. 87 and 88, the compounding curve of a direct-current generator. The arrange- V olts -550 ^ ■^ ,^ ^ y y JLXt\ / 0 2 0 i 0 6 0 8 1 0 • Fig. 88. Compounding Curve. 258 ENGINEERING MATHEMATICS. ment in Fig. 87 is correct ; it shows the relative variation of voltage as function of the load. Fig. 88, in which the cross-sectioning does not begin at the scale zero, confers the — ___ __^ -- /^ ^ / / 9. ; / c K 1 3 2 > 3 ) * ) 5 3 6 3 7 D Fig. 89. Curve Plotted to show Characteristic Shape. Fig. 90. Curve Plotted for Use as De.sign Data. wrong impression that the variation of voltage is far greater than it really is. When curves are used to record numerical values and derive them from the curve, as, for instance, is connnonly the NUMERICAL CALCULATIONS. 259 case with magnetization curves, it is unnecessary to have the zero of the function coincide with the zero of the cross-sectioning, but rather preferable not to have it so, if thereby a better scale of the curve can be secured. It is desirable, however, to use suffi- ciently small cross-sectioning to make it possible to take numerical values from the curve with good accuracy. This is illustrated by Figs. 89 and 90. Both show the magnetic charac- teristic of soft steel, for the range above (B = 8000, in which it is usually employed. Fig. 89 shows the proper way of plotting for showing the shape of the function, Fig. 90 the proper way of plotting for use of the curve to derive numerical values therefrom. \ V \, \ \ I \ \ ^! \ \, \ k\ \ \ ^ s >s. ^ X ~^, III ^ -- '~~ ~ Fig. 91. Same Function Plotted to Different Scales; I is correct. 165. Curves should be plotted in such a manner as to show the quantity which they represent, and its variation, as well as possible. Two features are desirable herefor: 1. To use such a scale that the average slope of the curve, or at least of the more important part of it, does not differ much from 45 deg. Hereby variations of curvature are best shown. To illustrate this, the exponential function y = e~'^ is plotted in three different scales, as curves I, II, III, in Fig. 91. Curve I has the proper scale. 2. To use such a scale, that the total range of ordinates is not much different from the total range of abscissas. Thus when plotting the power-factor of an induction motor, in Fig. 92, curve I is preferable to curves II or III. 260 ENGINEERING MATHEMATICS, These two requirements frequently are at variance with each other, and then a compromise has to be made between them, that is, such a scale chosen that the total ranges of the two coordinates do not differ much, and at the same time the average slope of the curve is not far from 45 deg. This usually leads to a somewhat rectangular area covered by the curve, as shown, for instance, by curve I, in Fig. 91. In curve plotting, a scale should be used which is easily read. Hence, only full scale, double scale, and half scale should be used. Triple scale and one-third scale are practically unreadable, and should therefore never be used. Quadruple 7 ^ "^ ^ T- ^ / / / y' / / / 1 / / III ^ — r ^ ^ / ^ Y / Fig. 92. Same Function Plotted to Different Scales; I is Correct. scale and quarter scale are difficult to read and therefore unde- sirable, and are generally unnecessary, since quadruple scale is not much different from half scale with a ten times smaller unit, and quarter scale not much different from double scale of a ten times larger unit. i66. Any engineering calculation on which it is worth while to devote any time, is worth being recorded wdth suffi- cient completeness to be generally intelligible. Very often in making calculations the data on which the calculation is based, the subject and the purpose of the calculation are given incom- pletely or not at all, since they are familiar to the calculator at the time of calculation. The calculation thus would be unin- NUMERICAL CALCULATIONS. 261 telligible to any other engineer, and usually becomes unintelli- gible even to the calculator in a few weeks. In addition to the name and the date, all calculations should be accompanied by a complete record of the object and purpose of the calculation, the apparatus, the assumptions made, the data used, reference to other calculations or data employed, etc., in shorty they should include all the information required to make the calculation intelligible to another engineer without further information besides that contained in the calculations, or in the references given therein. The small amount of time and work required to do this is negligible compared with the increased utility of the calcuktion. Tables and curves belonging to the calculation should in the same way be completely identified with it and contain sufficient data to be intelligible. d. Reliability of Numerical Calculations. 167. The most important and essential requirement of numerical engineering calculations is their absolute reliability. When making a calculation, the most brilliant ability, theo- retical knowledge and practical experience of an engineer are made useless, and even worse than useless, by a single error in an important calculation. Reliability of the numerical calculation is of vastly greater importance in engineering than in any other field. In pure mathematics an error in the numerical calculation of an example which illustrates a general proposition, does not detract from the interest and value of the latter, which is the main purpose; in physics, the general law which is the subject of the' investigation remains true, and the investigation of interest and use, even if in the numerical illustration of the law an error is made. With the most brilliant engineering design, however, if in the numerical calculation of a single structural member an error has been made, and its strength thereby calcu- lated wrong, the rotor of the machine flies to pieces by centrifugal forces, or the bridge collapses, and with it the reputation of the engineer. The essential difference between engineering and purely scientific caclulations is the rapid check on the correct- ness of the calculation, which is usually afforded by the per- 262 ENGINEERING MATHEMATICS. forniance of the calculated structure — but too late to correct errors. Thus rapidity of calculation, while by itself useful, is of no value whatever compared with reliability — ^that is, correct- ness. One of the first and most important requirements to secure reliability is neatness and care in the execution of the calcula- tion. If the calculation is made on any kind of a sheet of paper, with lead pencil, with frequent striking out and correct- ing of figures, etc., it is practically hopeless to expect correct results from any more extensive calculations. Thus the work should be done with pen and ink, on white ruled paper; if changes have to be made, they should preferably be made by erasing, and not by striking out. In general, the appearance of the work is one of the best indications of its reUability. The arrangement in tabular form, where a series of values are calcu- lated, offers considerable assistance in improving the reliability. i68. Essential in all extensive calculations is a complete system of checking the results, to insure correctness. One way is to have the same calculation made independently by two different calculators, and then compare the results. Another way is to have a few points of the calculation checked by somebody else. Neither way is satisfactory, as it is not always possible for an engineer to have the assistance of another engineer to check his work, and besides this, an engineer should and must be able to make numerical calculations so that he can absolutely rely on their correctness without somebody else assisting him. In any more important calculations every operation thus should be performed twice, preferably in a different manner. Thus, when multiplying or dividing by the slide rule, the multi- plication or division should be repeated mentally, approxi- mately, as check; when adding a column of figures, it should be added first downward, then as check upward, etc. Where an exact calculation is requked, first the magnitude of the quantity should be estimated, if not already known, then an approximate calculation made, which can frequently be done mentally, and then the exact calculation; or, inversely, after the exact calculation, the result may be checked by an approximate mental calculation. NUMERICAL CALCULATIONS. 2G3 Where a series of values is to be calculated, it is advisable first to calculate a few individual points, and then, entirely independently, calculate in tabular form the series of values, and then use the previously calculated values as check. Or, inversely, after calculating the series of values a few points should independently be calculated as check. When a series of values is calculated, it is usually easier to secure rehability than when calculating a single value, since in the former case the different values check each other. There- fore it is always advisable to calculate a number of values, that is, a short curve branch, even if only a single point is required. After calculating a series of values, they are plotted as a curve to see whether they give a smooth curve. If the entire curve is irregular, the calculation should be thrown away, and the entire work done anew, and if this happens repeatedly with the same calculator, the calculator is advised to find another position more in agreement with his mental capacity. If a single point of the curve appears irregular, this points to an error in its calculation, and the calculation of the point is checked; it the error is not found, this point is calculated entirely separately, since it is much more difficult to find an error which has been made than it is to avoid making an error. 169. Some of the most frequent numerical errors are : 1. The decimal error, that is, a misplaced decimal point. This should not be possible in the final result, since the magni- tude of the latter should by judgment or approximate calcula- tion be known sufficiently to exclude a mistake by a factor 10. However, under a square root or higher root, in the exponent of a decreasing exponential function, etc., a decimal error may occur without affecting the result so much as to be immediately noticed. The same is the case if the decimal error occurs in a term which is relatively small compared with the other terms, and thereby does not affect the result very much. For instance, in the calculation of the induction motor characteristics, the quantity ri^+s^xi^ appears, and for small values of the sHp s, the second term s^xi^ is small compared with ri^, so that a decimal error in it would affect the total value sufficiently to make it seriously wrong, but not sufficiently to be obvious. 2. Omission of the factor or divisor 2. 264 ENGINEERING MATHEMATICS. 3. Error in the sign, that is, using the plus sign instead of the minus sign, and inversely. Here again, the danger is especially great, if the quantity on which the wrong sign is used combines with a larger quantity, and so does not affect the result sufl^ciently to become obvious. 4. Omitting entire terms of smaller magnitude, etc. APPENDIX A. NOTES ON THE THEORY OF FUNCTIONS. A. General Functions. 170. The most general algebraic expression of powers of X and y, F(x,y) = (aoo+aoiX-\-ao2X^ + . . .) +(^10 + «i 1:^^ + ^112.^^ + . . .)2/ + {a2o+a2ix + a22X^ + . . .)2/^ + - • • + {ano+anix+an2X^ + . . .)y''-0, . . . . (1) is the implicit analytic function. It relates y and x so that to every value of x there correspond n values of y, and to every value of y there correspond m values of x, if m is the exponent of the highest power of a: in (1). Assuming expression (1) solved for y (which usually cannot be carried out in final form, as it requires the solution of an equation of the r^th order in y, with coefficients which are expressions of x), the explicit analytic function, y-m, (2) is obtained. Inversely, solving the implicit function (1) for X, that is, from the explicit function (2), expressing x as function of y, gives the reverse function of (2); that is x=/i(2/) (3) In the general algebraic function, in its implicit form (1), or the explicit form (2), or the reverse function (3), x and y are assumed as general numbers; that is, as complex quan- tities; thus, X = Xi+jX2] \ (4) y=yi+jy2, J and likewise are the coefficients ooo, aoi • • . cinn'. 265 266 ENGINEERING MATHEMATICS. If all the coefficients a are real, and x is real, the corre- sponding n values of y are either real, or pairs of conjugate complex imaginary quantities: 2/1 +^2/2 and y\ — jy2. 171. For 71 = 1, the implicit function (1), solved for y, gives the rational function, aoo+aoiX + ao2y^ + . . . , . and if in this function (5) the denominator contains no x, the integer function, y = ai)+aix+a2X^-\ . . .-\-amX'^, , . . . (6) is obtained. For n = 2, the implicit function (1) can be solved for 1/ as a quadratic equation, and thereby gives -{aio+anX + ai-2X^ + ...)± that is, the explicit form (2) of equation (1) contains in this case a square root. For n>2, the explicit form y=f{x) either becomes very complicated, for n = 3 and n = 4, or cannot be produced in finite form, as it requires the solution of an equation of more than the fourth order. Nevertheless, y is still a function of x, and can as such be calculated by approximation, etc. To find the value yi, which by function (1) corresponds to x^x\, Taylor's theorem offers a rapid approximation. Sub- stituting xi in function (1) gives an expression which is of the nth order in y, thus: F(xiy), and the problem now is to find a value y\, which makes F(xi,yi) =0. However, X r./ X ,dF(xi,y) h^d^F{xi,7j) Fix,, yi)==F(x,, y) +^— ^- +j2 ^y ' +- ^ , • i^) where h = yi — y is the difference between the correct value t/i and any chosen value y. APPENDIX A, 267 Neglecting the higher orders of the small quantity h, in (8), and considering that F(xi,yi) =0, gives h = F{x,,y) dF{x^,yy dy (9) and herefrom is obtained yi=y-\-h, as first approximation. Using this value of i/i as y in (9) gives a second approximation, which usually is sufficiently close. 172. New functions are defined by the integrals of the analytic functions (1) or (2), and by their reverse functions. They are called Ahelian integrals and Abelian functions. Thus in the most general case (1), the explicit function corresponding to (1) being the integral, z= i f{x)dx, (2) then is the general Abelian integral, and its reverse function, x = {z), the general Abelian function. (a) In the case, n = l, function (2) gives the rational function (5), and its special case, the integer function (6). Function (6) can be integrated by powers of x. (5) can be resolved into partial fractions, and thereby leads to integrals of the following forms : (1) I xmdx] (2) P-; J x-a' (4) r dx J 1 +^' (10) 268 ENGINEERING MATHEMATICS. Integrals (10), (1), and (3) integrated give rational functions, (10), (2) gives the logarithmic function log (x—a), and (10), (4) the arc function arc tan x. As the arc functions are logarithmic functions with complex imaginary argument, this case of the integral of the rational function thus leads to the logarithmic function, or the loga- rithmic integral, which in its simplest form is z-^l -^ = logx, ....... (11) and gives as its reverse function the exponential function, X=£^ (12) It is expressed by the infinite series, ^2 ^3 ^ e^=l+e+_+_+jj+ . (13) as seen in Chapter II, paragraph 53. 173. b. In the case, n=2, function (2) appears as the expres- sion (7), which contams a square root of some power of x. It.s first part is a rational function, and as such has already been discussed in a. There remains thus the integral function, /Vbo+hiX + b2X^-\-. . .+6p.i:^^ ■ ■ 5-^ dx. . . . (14) Co+ClX-{-C2X^-\-. . . This expression (14) leads to a series of important functions. (1) Forp = lor2, r Vbo+biX + b2X^ ^ ^_. Z= I ; : ^- dx (15) J Co-\-CiX+C2X^-\-. . • By substitution, resolution into partial fractions, and separation of rational functions, this integral (11) can be reduced to the standard form, dx In the case of the minus sign, this gives '-jr^^^ ^^^^ /dx z= I — = arc sin x, (17) APPENDIX A. 2m and as reverse functions thereof, there are obtained the trigo- notrteiric functions. x = sin z, 1 (IS) Vl — X^ = COS z. J In the case of the plus sign, integral (16) gives = — log{ Vl+x^— xj =arc sinh x, . Vl+X2 and reverse functions thereof are the hyperbolic functions, c + 2— e-Z 1 (19) X vTTx2= 2 sinh^;; = cosh z. (20) The trigonometric functions are expressed by the series : y3 9;5 y7 sin2 = 2-Tj + l^-|y + .: . ; ^2 ^4 ^ COS^=l-^+^-jg+..., (21) as seen in Chapter II, paragraph 58. The hyperbolic functions, by substituting for £+« and e' the series (13), can be expressed by the series: z ?^ z"^ sinh z = z+77^+^-\-T=- + . . . ; |3 |o \!_ z^ ^ ifi coshe=l+i2+jj+jg+.... (22) 174. In the next case, p=3 or 4, /Vfeo +61X +620:2 +63.^^ + ?>4^ Co + CiX+C2X2 + dx, (23) already leads beyond the elementary functions, that is, (23) cannot be integrated by rational, logarithmic or arc functions, 270 ENGINEERING MATHEMATICS. but gives a new class of functions; the elliptic integrals, and their reverse functions, the elliptic functions, so called, because they bear to the eUipse a relation similar to that, which the trigonometric functions bear to the circle and the hyperbolic functions to the equilateral hyperbola. The integral (23) can be resolved into elementary functions, and the three classes of elliptic integrals : dx vx{i-x){i-cHy xdx Vx{i-x){i-cHy dx h)Vx(l-x)(l-c^x) (24) (These three classes of integrals may be expressed in several different forms.) The reverse functions of the elliptic integrals are given by the elliptic functions : Vx = sin am(u, c); V 1 — x = cos am{u, c) ; Vl —c^x = Jam{u, c); (25) known, respectively, as sine-amplitude, cosine-amplitude, delta- amplitude. Elliptic functions are in some respects similar to trigo- nometric functions, as is seen, but they are more general, depending, as they do, not only on the variable x, but also on the constant c. They have the interesting property of being doubly periodic. The trigonometric functions are periodic, with the periodicity 2;r, that is, repeat the same values after every change of the angle by 2;r. The elliptic functions have two periods pi and p2, that is, sin am{u +npi +w,p2, c) =sin am{u, c), etc.; (26) hence, increasing the variable u by any multiple of either period pi and p2, repeats the same values. APPENDIX A, 271 The two periods are given by the equations, dx Vi -X' 2\^x{\-x){l-cHy dx (27) 2\/x{l-x){l-c^x) ] 175. Elliptic functions can be expressed as ratios of two infinite series, and these series, which form the numerator and the denominator of the elliptic function, are called theta func- tions and expressed by the symbol 6, thus sinaw(u, c)=-^ /-..A' COS am(u, c) =y\-i _ 2 ».© Jam{u, c) == -yll — c^- (28) '-is)' and the four 6 functions may be expressed by the series: ^o(^) = 1 —2q cos 2^ +2^^ cos 4a: —2q^ cos 6a; H — . . . ; 25 di(x) =2gi/4 sin x -2g9/4 sin 3x +2g4 sin 5a; - + . . . ; 25 ^2^=2(2^/4 cos a; +2^9/4 COS 3a: +2^ 4 cos5a: + ... ; (93(x) = l+2(?cos2a:+2^cos4a:+259cos6a: + . . . , (29) where 1 . P2 q=s9- and a = '\n^—. (30) In the case of integral function (14), where p>4, similar integrals and their reverse functions appear, more complex 272 ENGINEERING MATHEMATICS. than the elliptic functions, and of a greater number of periodici- ties. They are called hyperelliptic integrals and hyperelliptic functions, and the latter are again expressed by means of auxil- iary^ functions, the hyperelliptic 6 functions. 176. Many problems of physics and of engineering lead to elliptic functions, and these functions thus are of considerable^ importance. For instance, the motion of the pendulum is expressed by elliptic functions of time, and its period thereby is a function of the amplitude, increasing with increasing ampli- tude: that is, in the so-called "second pendulum," the time of one swing is not constant and equal to one second, but only approximately so. This approximation is very close, as long as the amplitude of the swing is very small and constant, but if the amplitude of the swing of the pendulum varies and reaches large values, the time of the swing, or the period ot the pendulum, can no longer be assumed as constant and an exact calculation of the motion of the pendulum by elliptic functions becomes necessary. In electrical engineering, one has frequently to deal w^ith oscillations similar to those of the pendulum, for instance, in the hunting or surging of synchronous machines. In general, the frequency of oscillation is assumed as constant, but where, as in cumulative hunting of synchronous machines, the amplitude of the swing reaches large values, an appreciable change of the period must be expected, and where the hunting is a resonance effect with some other periodic motion, as the engine rotation, the change of frequency with increase of amplitude of the oscillation breaks the complete resonance and thereby tends to limit the amplitude of the swing. 177. As example of the application of elliptic integrals, may be considered the determination of the length of the arc of an ellipse. Let the ellipse of equation 5-^4=1' (31) be represented in Fig. 93, with the circumscribed circle, jc2 + y2 = a2 (32) APPENDIX A. 273 To every point P = x, y of the ellipse then corresponds a point P\ = x, y\ on the circle, which has the same abscissa x, and an angle d^AOP\. The arc of the ellipse, from A to P, then is given by the integral, L = a \ — , ^ ^ =zi .... (33) where z){l-c'z) z = ^m^ d=(-] and c is the eccentricity of the ellipse. Va^ — b^ (34) Fig. 93. Rectification of Ellipse. Thus the problem leads to an elliptic integral of the first and of the second class. For more complete discussion of the elliptic integrals and the elliptic functions, reierence must be made to the text-books of mathematics. B. Special Functions. 178. Numerous special functions have been derived by the exigencies of mathematical problems, mainly of astronomy, but in the latter decades also of physics and of engineering. Some of them have already been discussed as special cases of the general Abelian integral and its reverse function, as the expo- nential, trigonometric, hyperbolic, etc., functions. 274 ENGINEERING MATHEMATICS. Functions may be represented by an infinite series of terms; that is, as a ^um of an infinite number of terms, which pro- gressively decrease, that is, approach zero. The denotation of the terms is commonly represented by the summation sign 2. Thus the exponential functions may be written, when defining, |0 = 1; |n = lX2X3X4X. . .Xn, as e- = l+x+;^*+j^+. ..==S»;--, .... (35) I— l_. ^ l__ which means, that terms ,— are to be added for all values of n from n = 0 to n = oc . The trigonometric and hyperbolic functions may be written in the form : sina:-x--~+T^-j;^ + . . . = Sn(-i)n • . (36) j3 |5 |7 0 ' ' |2n + l .y2 7^ /y.6 oo ^2n cosx=l-|^+ij-g + ... = 2»r-l)ni^; . . (37) x^ x^ x"^ ^ X^""*"^ sinhx-a:+j^+y^+yy+. . . = 2n-^— -^; . . . (38) eosha. = l+g+^+i^ + ... = Snj^ (39) Functions also may be expressed by a series of factors; that is, as a product of an infinite series of factors, which pro- gressively approach unity. The product series is commonly represented by the symbol JJ. Thus, for instance, the sine function can be expressed in the form, sinx=x(l-5)(l-£,)(l-£)... = xt(l-i). (40) 179. Integration of known functions frequently leads to new functions. Thus from the general algebraic functions were APPENDIX A. 275 derived the Abelian functions. In physics and in engineering, integration of special functions in this manner frequently leads to new special functions. For instance, in the study of the propagation through space, of the magnetic field of a conductor, in wireless telegraphy, lightning protection, etc., we get new functions. If ^=/(0 is the current in the conductor, as function of the time t, at a distance x from the conductor the magnetic field lags by the X time ti = -, where S is the speed of propagation (velocity of light). Since the field intensity decreases inversely propor- tional to the distance x, it thus is proportional to y= — - — ; (41) and the total magnetic flux then is / 2= j ydx A'-l) -j^T^'i' <*2) If the current is an alternating current, that is, f (t) a trigonometric function of time, equation (42) leads to the functions, /sin ; X "J dx: cos X ^ -ax. (43) If the current is a dh-ect current, rising as exponential function of the time, equation (42) leads to the function, /s'^dx --•••' (44) X 27G ENGINEERING MATHEMATICS. Substituting in (43) and (44), for sin x, cos a:, £^ their infinite series (21) and (13), and then integrating, gives the following : sin X x^ x^ x^ X '^*^^^~3]3"^5J5"7|7^~ J sin X r cosa; x^ X* x^ ~rfx = logx-2J2+4J4-gje (15) ^dx = logx+x+2|+3|3+... For further discussion of these functions see 'Theory and Calculation of Transient Electric Phenomena and Oscillations," Section III, Chapter VIII. i8o. If y=f{x) is a function of x, and z= | f {x)dx = 6{x) its integral, the definite integral, ^^ \ f{^)dx, is no longer Ja a function of x but a constant, For instance, if y^c{x — nY, then c{x—n)^ 0= I c{x—n )Hx = and the definite integral is =r^^'- nYdx = -\{h — n)^—{a—n)^\. This definite integral does not contain x, but it contains all the constants of the function / {x), thus is a function of these constants c and n, as it varies with a variation of these constants. In this manner new functions may be derived by definite integrals. Thus, if y=f{x,u,v...) (46) is a function of x, containing the constants u, v , . , APPENDIX A. 277 The definite integral, , . - . Z=rf(x,u,v...)dx, . .... (47) is not .a function of x, but still is a function o( u, v . . . , and may be a new function. i8i. For instance, let " y^s-^x""-^; (48) then the integral, /(u)= n--x^-Mx, (49) is a new function of u, called the gamma function. Some properties of this function may be derived by partial integration, thus : r(u + l)=ur(u); (50) if n is an integer number, r(u) = (u-l){u-2)...(u-n)r(u-n), . . (51) and since r(i)=i, (52) if u is an integer number, then, r(u) = \u-h (53) C. Exponential, Trigonometric and Hyperbolic Functions. (a) Functions of Real Variables. 182. The exponential, trigonometric, and hyperbolic func- tions are defined as the reverse functions of the integrals, a. u= j— =logx, (54) and: x=£« (55) ...... (56) r dx h. u= I y ==arcsmx; 278 ENGINEERING MATHEMATICS. and: x=8mu, (57) Vl-x^ = cosu, (58) /dx ;^^^^===-log{Vr+^-xl; . . . . (59) and x= X — =sinhw; .... (60) Vl+x2= — =coshw (61) From (57) and (58) it follows that sm^u+cos^u = l (62) From (60) and (61) it follows that cos^/ii/ — sin 2/ii^ = l (63) Substituting ( — x) for x in (56), gives {—u) instead of u, and therefrom, sin (— w) = — sin 1^ (64) Substituting {—u) for u in (60), reverses the sign of x, that is, sinh ( — w) = — sinhii. ... (65) Substituting {—x) for x in (58) and (61), does not change the value of the square root, that is, cos ( — w)=cos w, (66) cosh {—u)= cosh u, (67) Which signifies that cos u and cosh u are even functions, while sin u and sinh u are odd functions. Adding and subtractijig (60) and (61), gives £±M = cosh w± sinh ?/ (68) APPENDIX A. 279 (h) Functions of Imaginary Variables. 183. Substituting, in (56) and (59), a: = —jy, thus y = jx, gives dx ^^^'^ ^=/vfo^ (^^-^ ^=/ Vl+x^' a: = sinw; x=sinhi^ = \/l+x^ = cosu; vTTx2 = coshu = 2 ' hence, y,,= J-^, hence, ju=j'^^; 2/=sinhfu= ^5 ; 2/=sin/w; . . . (69) £JU _|_ g-JU V 1+2/^ = cosh /i^ = 2 5 vl — 2/2 = cos/i^; . . . (70) Resubstituting x in both sinh/u £jw-£-i« £" — £-" sinm ,^,, ic = smw = ^ — = — ?P ; a: = sinhi^ = — t: — = — :^—: (71) ] 2? '- 2 J Vl — x2 = cos w = cosh ju Vi+x2 = cosht^ = £«+£-^ 2 =cos/w. . (72) Adding and subtracting, £^" = cos u±j sin w=cosh /w±sinh /i^ and £ ± « = cosh i* ± sinh u = cos /w T J sin jw. . . (73) (c) Functions of Complex Variables 184. It is : £u±jt,= £^e±/t»=£u(cosi;±/smt;); . . . (74) 280 ENGINEERING MATHEMATICS. sin (u i/iO =sin u cos ;v ±cos u sin jv = Sin w cosh V ± ^ cos w sinh v = — ^ — sin u ± ] — ^ — cos w cos(w ± jv) = cos It cos p^ sin w sin jv = cos u cosh ?; ^ /sin u sinh r = — ^ — cos u =F j — ^ — sin u ; (75) ■ (76) sinh {u ± jv) = ^ = — ^ — cos v ± j — ^ — sin v = sinh u cos v±j cosh u sin v\ .U ff.— u ^ 2 = cosh u cos V ±y sinh ti sin v; cosh(2^ ± jv) = ~ = ^^ — '-^ — cos V ± j — ^ — sin v etc. (77) \ (78) (d) Relations. 185. From the preceding equations it thus follows that the three functions, exponential, trigonometric, and hyperbolic, are so related to each other, that any one of them can be expressed by any other one, so- that when allowing imaginary and complex imaginary variables, one function is sufficient. As such, naturally, the exponential function would generally be chosen. Furthermore, it follows, that all functions with imaginary and complex imaginary variables can be reduced to functions of real variables by the use of only two of the three classes of functions. In this case, the exponential and the trigono- metric functions would usually be chosen. Since functions with complex imaginary variables can be resolved into functions with real variables, for their calculation tables of the functions of real variables are sufficient. The relations, by which any function can be expressed by any other, are calculated from the preceding paragraph, by the following equations : APPENDIX A. 281 f ±" = cosh u ±sinh u = cos fu ^F / sin fu; £'^^^' = coi>v± j sin V = cosh jv ± j sinh jv ; £"±/'' == £« (cos v±j sin i;), sinh ju £^""— £~^'" sni w 2/ ' sin jv = j sinh r = / — ^ sin {u±jii) =sin i^ cosh ?? ±/ cos u sinh v = — ^ — sin u±j — - — cos u ; (a) (&) COS u = cosh ju cos p = cosh V £?M^ C-7U cos {u±jv) = COS u cosh i;=F / sjn 21 sinh 2; = — ^^ — COS 2iTJ- — 7^ — sin u; (e) sinh ti = sinh jv = j sin 2 £"— £ " sin JU 2 7 ' sinh {u±jv) =sinh u cos ?; ±/ cosh w sin v -£-" .£«+£-" . 2 — cos vi; — 2 — ^^^ ^j £^— £ id) cosh w = £"+£-" cos ]U\ £?U -j- £-?v cosh jv = cos V = ^ — ; cosh (u ± jv) = cosh u cos v±j sinh u sin ?; £«-f£-W ffW_ £ — U = :^ cos v±j sin v. ie) 282 ENGINEERING MATHEMATICS. And from (b) and (d), respectively (c) and (e), it follows that sinh {u + jv) = j sin {±v— ju) = ±j sin {v ± ju) ; cosh ('U±/i')=cos {v^^ju). } . c^ Tables of the exponential functions and their logarithms, and of the hyperbolic functions with real variables, are given in the following Appendix B. APPENDIX B. TWO TABLES OF EXPONENTIAL AND HYPERBOLIC FUNCTIONS. Table I. 6 = 2.7183. log £ = 0.4343. X xio-» XlO-2 xio-i XI 1.0 0.999 0.990 0.905 0.368 1.2 1.4 1.6 1.8 0.988 0.986 0.984 0.982 0.887 0.869 0.852 0.835 0.301 0.247 0.202 0.165 2.0 0.998 0.980 0.819 0.135 2.5 3.0 3.5 4.0 4.5 0.997 0.996 0.975 0.970 0.966 0.961 0.956 0.779 0.741 0.705 0.670 0.638 0.082 0.050 0.030 0.018 0.011 5.0 0.995 0.951 0.607 0.007 6 7 8 9 0.994 0.993 0.992 0.991 0.942 0.932 0.923 0.914 0.549 0.497 0.449 0.407 0.002 0.001 0.000 10 0.990 0.905 0.368 283 284 ENGINEERING MATHEMATICS. Table IL EXPONENTIAL AND HYPERBOLIC FUNCTIONS. £ = 2.718282^2.7183, log £ = 0.4342945 ~ 0.4343. cosh x = h{£ + ^ + £-^\, sinhx = ^|£ + ^-£--^|. ,, ! 434 435 0 0 0 0.1 43 43 0.2 87 87 0.3 130 130 0.4 174 174 0.5 217 217 0.6 261 261 0.7 304 304 0.8, 347 348 0.9 391 391 1.0 434 435 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.014 0.016 0.018 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 logfH-x 0.000434 0.000869 0.001303 0.001737 0.002171 0.002606 0 . 003040 0.003474 0.003909 0.004343 0.005212 0.006080 0.006949 0.007817 0.008686 0.010857 0.013029 0.015200 0.017372 0.019543 0.02171i 0.026058 0.030401 0.034744 LO. 039086 0.043429 0.052115 0.060801 0.069487 0.078173 0.20 0.086859 Jlog £±^ 434 435 434 434 434 435 434 434 435 434 log £~^ .999566 1 .999131 00100 1.00200 1.00301 9.99826311.00401 9.997829 1.00501 9.997394 9.996960 1 9.996526 1 9.996091 9.995657 .994788 .993920 1 .993051 .992183 1 9.991314 9.989143 1 9.986971 9.984800 9.982628 1 9.98045' 9.978285 1.05127 (3942 1 9.97 9.969599 1 9.9652.56 1 9.960914 9.956571 9.947885 9.939199 9.930513 1 9.921827 1 £ + X 0.99900 1 0.99800 1 0.99700 1 0.99601 1 . 00602 00702 00803 1.00904 1.01005 1.01207 01410 1.01613 01816 1.02020 02531 1.03046 1.03562 .04081 1.04603 .06184 .07251 .08329 1.09417 1.10516 1 . 12750 1 . 15027 17351 19721 0.99501 0.9940: 0.99302 1 0.99203 1 0.99104 1 0.99005 1.00005 0.9880' 0.98610 1 0.98413 1 0.98216 1 0.97531 0.97045 1 0.96561 1 0.96079 1 0.95600 1 0.941 0 . 9323911 0.9231 0.91393 0.90484 0.8869 0.86936 0.85214 0.83527 9.913141 1.22140 0.81873 1.02006 0.20134 0.20 cosh X 00000 00000 00000 1.00001 1.00001 1.00002 .00002 .00003 . 00004 1.00007 .00010 .00013 .00016 0.98020 1.00020 00031 00046 00062 00080 00102 0.95123 1.00125 0.05003 sinh 0.00100 0.00200 0.00300 0.00400 0.00500 0.00600 0.00700 0.00800 0.00900 0.01000 0.01200 0.01400 0.01600 0.01800 0.02000 0.02500 0.03000 0.03500 0.04001 0.04502 6 1 00180 0.06004 00245 0.07006 1. 00321; 0.08008 1.00405 0.09011 1.00500 0.10016 i 1.00721 0.12028 1.00982 0.14046 1.01283 0.16069 1.01624 0.18097 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.014 0.016 0.018 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.06 0.07 0.08 0.09 0 . 10 0.12 0.14 0.16 0.18 £ + 0.001 = 1.001000494, f-o^O' = 0.99900049. APPENDIX B, Table II — Continued. EXPONENTIAL AND HYPERBOLIC FUNCTIONS. 285 X log e + ^ log £-^ £ + ^ £-^ cosh X sinh X X 0.20 0.086859 9.913141 1.22140 0.81873 1.02006 0.20134 0.20 0.25 0.30 0.35 0.40 0.45 0.108574 0.130288 0.152003 0.173718 0.195433 9.891426 9.869712 9.847997 9.826282 9.804567 1 . 28403 1.34986 1.41907 1.49183 1.56831 0.77880 0.74082 0.70469 0.67032 0.63763 1.03142 1.04534 1.06188 1.08108 1 . 10297 0.25261 0.30457 0.35719 0.41076 0.46534 0.25 0.30 0.35 0.40 0.45 0.50 0.217147 9.782853 1.64870 0.60653 1.12761 0.52108 0.50 0.6 0.7 0.8 0.9 0.260577 0.304006 0.347436 0.390865 9.739423 9.695994 9.652564 9.609135 1.82212 2.01375 2.22554 2.45960 0.54881 0.49659 0.44933 0.40657 1 . 19546 1.25517 1.33744 1.43309 0.63666 0.75858 0.88811 1.02657 0.6 0.7 0.8 0.9 1.0 0.434294 9.565706 2.71828 0.36788 1.54308 1 . 17520 1.0 1.2 1.4 1.6 1.8 0.521153 0.608012 0.694871 0.781730 9.478847 9.391988 9.305129 9.218270 3.32011 4.05520 4.95304 6.04965 0.30119 0.24660 0.20190 0.16530 1.81065 2.15090 2.57745 3.10745 1.50946 1.90430 2.37557 3.44218 1.2 1.4 1.6 1.8 2.0 0.868589 9.131411 7.38906 0.13534 3.76220 3.62686 2.0 2.5 3.0 3.5 4.0 4.5 1.085736 1.302883 1.520030 1.737178 1.954325 8.914264 8.694117 8.479970 8.262822 8.045675 12.1825 20.0855 33.1154 54.5983 90.0170 0.082085 0.049797 0.030197 0.018316 0.011109 6.1323 10.0677 16.5728 27.3083 45.0141 6.0002 10.0178 16.5426 27 . 2900 45.0030 2.5 3.0 3.5 4.0 4.5 5.0 2.171472 7.828528 148.413 0.006738 74.210 74.203 5.0 6 7 8 9 2.605767 3.040061 3.474356 3.908650 7.394233 6.959939 6.525644 6.091350 403.428 1096.63 2980.96 8103.08 0.002479 0.000912 0.000335 0.000123 201.715 201.713 6 7 8 9 10 for X >6 10 4.342945 5.657055 22026.5 0.0000454 12 14 16 18 5.211534 6.080123 6.948712 7.817301 4.788466 3.919877 3.051288 2.182699 162755 1202610 8886120 65660000 0.0000061 0.00000083 0.00000011 0.00000002 12 14 16 18 20 8.685890 1.314110 485166000 0.00000000 20