A TREATISE ON BESSEL FUNCTIONS AND THEIR APPLICATIONS TO PHYSICS. 0 0 0 A TREATISE ON BESSEL FUNCTIONS AND THEIR APPLICATIONS TO PHYSICS. BY ANDREW GRAY, M.A., PROFESSOR OF PHYSICS IN THE UNIVERSITY COLLEGE OF NORTH WALES AND G. B. MATHEWS, M.A., FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY COLLEGE OF NORTH WALES. "And as for the Mixt Mathematikes I may onely make this prediction, that there cannot faile to bee more kindes of them, as Nature growes furder disclosed." BACON. Lontron: MACMILLAN AND CO. AND NEW YORK. 1895 [All Rights reserved.] CIambrtfige: PRINTED BY J. & C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. THIS book has been written in view of the great and growing importance of the Bessel functions in almost every branch of mathematical physics; and its principal object is to supply in a convenient form so much of the theory of the functions as is necessary for their practical application, and to illustrate their use by a selection of physical problems, worked out in some detail. Some readers may be inclined to think that the earlier chapters contain a needless amount of tedious analysis; but it must be remembered that the properties of the Bessel functions are not without an interest of their own on purely mathematical grounds, and that they afford excellent illustrations of the more recent theory of differential equations, and of the theory of a complex variable. And even from the purely physical point of view it is impossible to say that an analytical formula is useless for practical purposes; it may be so now, but experience has repeatedly shown that the most abstract analysis may unexpectedly prove to be of the highest importance in mathematical physics. As a matter of fact it will be found that little, if any, of the analytical theory included in the present work has failed to be of some use or other in the later chapters; and we are so far from thinking that anything superfluous has been inserted, that we could almost wish that space would have allowed of a more extended treatment, especially in the chapters on the complex theory and on definite integrals. With regard to that part of the book which deals with physical applications, our aim has been to avoid, on the one hand, waste of vi PREFACE. time and space in the discussion of trivialities, and, on the other, any pretension of writing an elaborate physical treatise. We have endeavoured to choose problems of real importance which naturally require the use of the Bessel functions, and to treat them in considerable detail, so as to bring out clearly the direct physical significance of the analysis employed. One result of this course has been that the chapter on diffraction is proportionately rather long; but we hope that this section may attract more general attention in this country to the valuable and interesting results contained in Lommel's memoirs, from which the substance of that chapter is mainly derived. It is with much pleasure that we acknowledge the help and encouragement we have received while composing this treatise. We are indebted to Lord Kelvin and Professor J. J. Thomson for permission to make free use of their researches on fluid motion and electrical oscillations respectively; to Professor A. Lodge for copies of the British Association tables from which our tables IV., V., VI., have been extracted; and to the Berlin Academy of Sciences and Dr Meissel for permission to reprint the tables of J0 and J1 which appeared in the Abhandlungen for 1888. Dr Meissel has also very generously placed at our disposal the materials for Tables II. and III., the former in manuscript; and Professor J. McMahon has very kindly communicated to us his formulae for the roots of J, (x)= 0 and other transcendental equations. Our thanks are also especially due to Mr G. A. Gibson, M.A., for his care in reading the proof sheets. Finally we wish to acknowledge our sense of the accuracy with which the text has been set up in type by the workmen of the Cambridge University Press. The bibliographical list on pp. 289-291 must not be regarded as anything but a list of treatises and memoirs which have been consulted during the composition of this work. CONTENTS. CHAPTER I. INTRODUCTORY. BERNOULLI'S problem of the oscillating chain, 1; conduction of heat in a solid cylinder, 2; Bessel's astronomical problem, 3; Bessel's differential equation, 5. CHAPTER II. SOLUTION OF THE DIFFERENTIAL EQUATION. Solution by series when n is not an integer, 7; and when n is an integer or zero, 8; definition of J,, (x), 11; elementary properties, 13; definition and explicit expression of Y (x), 14; expressions for J_,(x) and Y (x) as integrals, 15. CHAPTER III. FUNCTIONS OF INTEGRAL ORDER. EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. X +00 Proof of the theorem exp - (t-t-l)= E J (.V) tn, with corollaries, 17; 2 -o Bessel's expressions of Jn (x) as a definite integral, 18; transformation of power-series into series of Bessel functions, 19; Neumann's expression for Y,, 23; the addition theorem, 24; Neumann's extension thereof, 25; expansions in series of squares and products of Bessel functions, 29; Schlomilch's theorem, 30. CHAPTER IV. SEMICONVERGENT EXPANSIONS. Solution of Bessel's equation by successive approximation, 34; new expression for J, (x) as a definite integral, 38; the semiconvergent series for J, (x) and Y, (x), 40; numerical value of log 2 -, 41; table of J7k+ (x), 42. viii CONTENTS. CHAPTER V. THE ZEROES OF THE BESSEL FUNCTIONS. Bessel's proof that J,,(x)=0 has an infinite number of real roots, 44; calculation of the roots, 46; Stokes's and McMahon's formulae, 49. [See also p. 241.] CHAPTER VI. FOURIER-BESSEL EXPANSIONS. Rayleigh's application of Green's theorem, 51; value of the integral ra Jn (Kr) J, (Xr) rdr, 53; application to the conduction of heat, 54; formulae for Fourier-Bessel expressions, 55, 56. CHAPTER VII. COMPLEX THEORY. Hankel's integrals, 59-65; the functions I,, IT,, 66-8. Lipschitz's proof of the semiconvergent series, 69. CHAPTER VIII. DEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS. CHAPTER IX. THE RELATION OF THE BESSEL FUNCTIONS TO SPHERICAL HARMONICS. CHAPTER X. VIBRATIONS OF MEMBRANES. Equation of motion, 95; solution by Bessel functions, 96; interpretation of the result, 97; case of an annular membrane, 99. CHAPTER XI. HYDRODYNAMICS. Rotational motion in a cylinder, 101; oscillations of a cylindrical vortex about a state of steady motion, 104; hollow irrotational vortex, 106; vortex surrounded by liquid moving irrotationally, 109; waves in a tank, 110; rotating basin, 113; motion of a viscous liquid, 116; Stokes's problem of the cylindrical pendulum, 118. CONTENTS. ix CHAPTER XII. STEADY FLOW OF ELECTRICITY OR OF HEAT IN UNIFORM ISOTROPIC MEDIA. Differential equations, 122; potential due to electrified circular disk, 124; flow of electricity from disk-source in infinite conducting medium, 126; flow in infinite medium between two parallel planes: problem of Nobili's rings, 128; flow in cylindrical conductor with source and sink at extremities of axis, 131; flow in infinite medium with one plane face separated from infinite plate electrode by film of slightly conducting material, 132; case in which the medium has another parallel plane face in which the second electrode is situated, 136; further limitation of conductor by cylindrical surface, 139; flow in cylindrical conductor with electrodes on the same generating line, 139. CHAPTER XIII. PROPAGATION OF ELECTROMAGNETIC WAVES ALONG WIRES. Equations of electromagnetic field, 142; modification for symmetry round an axis, 144; solution of the equations, 146; solution adapted to long cylindrical conductor in insulating medium, 147; case of slow signalling along a cable, 150; electric and magnetic forces in this case, 151; expansion of xJo (x)/Jo (x) in ascending powers of x, 153; approximative values of Jo (%)/1J (7) and Go (r7)/IG (r) for large values of r, 155; case of more rapid oscillations, 157; effective resistance and effective self-inductance of cable, 157; Hertz's solution for a time-periodic electric doublet, 161; Hertz's solution for a wire, 162. CHAPTER XIV. DIFFRACTION OF LIGHT. Diffraction through a circular orifice, 165; calculation of intensity of illumination, 166; expression of intensity by Bessel functions, 167; discussion of Lommel's U, V functions and their relations to Bessel functions, 170; application of results to Fraunhofer's diffraction phenomena, 178; graphical expression of intensity, 179; application to Fresnel's more general case, 181; graphical discussion, 182; positions of maxima and minima of illumination, 184; diffraction when the orifice is replaced by an opaque circular disk, 192; positions of maxima and minima, 194; intensity due to infinite linear source, 198; Struve's discussion of this case, 201; application to determination of space-penetrating power of telescope, 205; diffraction through narrow slit with parallel edges, 208; expression of Fresnel's integrals in terms of Bessel functions, 209. X CONTENTS. CHAPTER XV. MISCELLANEOUS APPLICATIONS. Small vibrations of a gas, and variable flow of heat in a sphere, 212; stability of vertical wire, 215; torsional vibrations of a solid cylinder, 218; Bernoulli's problem for a chain of varying density, 221; differential equation reducible to Bessel's standard form, 222. NOTE ON THE SECOND SOLUTION OF BESSEL'S EQUATION WHICH VANISHES AT INFINITY...... 223 EXAMPLES......... 226 FORMULAE FOR ROOTS OF BESSEL AND RELATED FUNCTIONS. 241 EXPLANATION OF TABLES....... 245 TABLES........... 247 TABLE I. J (x), -J1 (), for x=0, '01,..., 15'5.. 247 TABLE II. J (x) for x=0, 1,..., 24, and all integral values of n for which Jn,(x) is not less than 10-18. 266 TABLE III. The first fifty roots of J1 (x)=0, with the corresponding maximum or minimum values of Jo (x) 280 TABLE IV. Jo(x ^i) for x=0, 2,..., 6. 281 TABLE V. I1 (x) for x=0, -01,..., 5'1.... 282 TABLE VI. I,,(x) for n=0, 1,..., 11, and x=0, 2,..., 6. 285 BIBLIOGRAPHY......... 289 GRAPH OF Jo AND J1. CORRIGENDUM. The analysis on p. 39 is invalid if n> ~. But it holds good if n lies between - 2 and 9, and the semiconvergent expression for J, satisfies (asymptotically) the relations (i6) and (19) on p. 13, so that the formula is applicable for all real values of n. CHAPTER I. INTRODUCTORY. BESSEL'S functions, like so many others, first presented themselves in connexion with physical investigations; it may be well, therefore, before entering upon a discussion of their properties, to give a brief account of the three independent problems which led to their introduction into analysis. The first of these is the problem of the small oscillations of a uniform heavy flexible chain, fixed at the upper end, and free at the lower, when it is slightly disturbed, in a vertical plane, from its position of stable equilibrium. It is assumed that each element of the string may be regarded as oscillating in a horizontal straight line. Then if in is the mass of the chain per unit of length, 1 the length of the chain, y the horizontal displacement, at time t, of an element of the chain whose distance from the point of suspension is x, and if T, T + dT are the tensions at the ends of the element, we find, by resolving horizontally, mdx dx dx dt2 T ddY dx, mdx dx dx or _ d dy\ or m d ---~ = d-x ddx) Now, to the degree of approximation we are adopting, T = mg (- x); d2y (1 d=y dy and hence dt2 g( 2 l- d If we write z for ( - x), and consider a mode of vibration for which y = zenti, u being a function of z, we shall have d2i du n2 z- +-+ — ~= 0. dz2 dz g G. M. 1 2 INTRODUCTORY. [I. Let us put K2 = n2/g, and assume a solution of the form = ao + a1z + az2 +...... =Ea.z' then z (2a2 + 3. 2. a,z +...... + (r + 1) rc.+lz'-l +......) +(a +2az. (r+ 1)...... ++ l) +......) + KC2 (ao + alz +...... + c,.zr +-......) = 0, and therefore a + Kc2ao = 0, 4a2 + tK2aCl = 0................... (r' + 1)2 ar+ + IC2a = 0; so that = ao (1-2 + 4- -.2.84..-.) 2 2 2 32.22 -. 32. = ao (K, ), say. The series Q( (K, z), as will be seen presently, is a special case of a Bessel function; it is absolutely convergent, and therefore arithmetically intelligible, for all finite values of K and z. The fact that the upper end of the chain is fixed is expressed by the condition (K, l)=o, which, when 1 is given, is a transcendental equation to find c, or, which comes to the same thing, n. In other words, the equation ( (K, 1) =0 expresses the influence of the physical data upon the periods of the normal vibrations of the type considered. It will be shown analytically hereafter that the equation b (K, 1) = 0 has always an infinite number of real roots; so that there will be an infinite number of possible normal vibrations. This may be thought intuitively evident, on account of the perfect flexibility of the chain; but arguments of this kind, however specious, are always untrustworthy, and in fact do not prove anything at all. The oscillations of a uniform chain were considered by Daniel Bernoulli and Euler (Comm. Act. Petr. tt. vi, vii, and Acta Acad. Petr. t. v.); the next appearance of a Bessel function is in Fourier's Theorie Analytique de la Chaleur (Chap. VI.) in connexion with the motion of heat in a solid cylinder. It is supposed that a circular cylinder of infinite length is heated in such a way that the temperature at any point within it I.] INTRODUCTORY. 3 depends only upon the distance of that point from the axis of the cylinder. The cylinder is then placed in a medium which is kept at zero temperature; and it is required to find the distribution of temperature in the cylinder after the lapse of a time t. Let v be the temperature, at time t, at a distance x from the axis: then v is a function of x and t. Take a portion of the cylinder of unit length, and consider that part of it which is bounded by cylindrical surfaces, coaxial with the given cylinder, and of radii x, x q- dx. If K is the conductivity of the cylinder, the excess of the amount of heat which enters the part considered above that which leaves it in the interval (t, t + dt) is a7 8v a2/x av dx/ dt; - K a. 2rz + K (2 x+ a ( ax ax; or, say, dH = 27rK (x - + dxdt. The volume of the part is 27rxdx, so that if D is the density, av and C the specific heat, the rise of temperature is dv = A dt, where CD. 27rxdx v- dt = dH. Hence, by comparison of the two values of dH, av (av I av CD-= K (x + - ). at ax2 xa a Fourier writes k for K/CD, and assumes v = ize-"t, u being a function of x only; this leads to the differential equation d2h 1 du n - - +- d + Vt= 0; 2 and now, if we put = g, we find there is a solution it=A g12 - g9 Y 1 22. 42 2 22. 4. 62 which is substantially the same function as that obtained by Bernoulli, except that we have _-gx2 instead of /c2Z. The boundary condition leads to a transcendental equation to find g; but this is not the place to consider the problem in detail. Bessel was originally led to the discovery of the functions which bear his name by the investigation of a problem connected with elliptic motion, which may be stated as follows. 1-2 4 INTRODUCTORY. [I. Let P be a point on an ellipse, of which AA' is the major axis, S a focus, and C the centre. Draw the ordinate NPQ meeting the auxiliary circle in Q, and join CQ, SP, SQ. A' [i- - A ^, ^ Then in the language of astronomy, the eccentric anomnaly of P is the number of radians in the angle ACQ, or, which is the same thing, it is >, where area of sector A CQ area of semicircle AQA' It is found convenient to introduce a quantity called the mean anomaly, defined by the relation area of elliptic sector A SP area of semi-ellipse APA' (By Kepler's second law of planetary motion,, is proportional to the time of passage from A to P, supposing that S is the centre of attraction.) Now by orthogonal projection area of ASP: area of APA' = area of ASQ: area of AQA' = (A CQ - CSQ): AQA' = (-a2 - ~ ea2 sin b): 27ra = ( - e sin o): 7r, where e is the eccentricity. Hence /u, e, b are connected by the relation, = - e sin b. 3 Moreover, if /u and 4 vary while e remains constant, -,a is a periodic function of / which vanishes at A and A'; that is, when,/ is a multiple of rr. We may therefore assume 00 > - IU = A Arsin r/, 4 1 and the coefficients A. are functions of e which have to be determined. INTRODUCTORY. 5 Differentiating 4 with respect to,/, we have XrA. cos r/ = - 1, dk and therefore, multiplying by cos ro/ and integrating, 7rrA2 = ( - -) cos r/J d/ fo df ) -= -d cos r/u d/. Now = 0 when k= 0, and = wr when a/ = 7r; so that by changing the independent variable from / to G, we obtain ~7rrA, = cos r/c do, o = cos r (- e sin 6) do, ~and A ~. ~2 j'r and Al. = 2 cos r() - e sin 4)) do, 5 which is Bessel's expression for A, as a definite integral. The function A, can be expressed in a series of positive powers of e, and the expansion may, in fact, be obtained directly from the integral. We shall not, however, follow up the investigation here, but merely show that A,. satisfies a linear differential equation which is analogous to those of Bernoulli and Fourier. Write x for e, and put 7rr I = A. =. cos r (( -x sin 4) do; du then, after partial integration of d with respect to 4, we find that d2u 1 du - dxt + dx _= _- r2 cos r (o - x sin S) do dW2 + x dx - r.2 rr + s c os cos r(- x sin )) dd r2 7r -rt - 2 |{(1 - x cos) - 1}cos r ( - x sin b) do = - r2 - r [sin r ( - x sin 0)]o + 2 Lt / 2 r\ - (r-2) uG; d2Ou n1 duL /- X or finally dx2 + - -+ - d 1 - 't = 0. 6 INTRODUCTORY. [I. If we put rx = z, this becomes d2u l du -(- r2 0 6 d- +-7- + 1-z -=0, 6 dZ2 +z cdz z2 and this is what is now considered to be the standard form of Bessel's equation. If in Fourier's equation, d2C 1 dd nat dz-, + + — = 0, dX2 x dx K/ /n we put x A/ -=, the transformed equation is d2u 1 duz dz + z d + u = 0, dZ2 Z dZ which is a special case of Bessel's standard form with r = 0. The differential equation is, for many reasons, the most convenient foundation upon which to base the theory of the functions; we shall therefore define a Bessel function to be a solution of the differential equation dZ -+ — + 1-2 - t = = 0. dx2 x dx c / In the general theory no restriction is placed upon the value of n; the most important case for physical applications is when i is zero or a positive integer. Moreover when n is integral the analytical theory presents some special features; so that for both reasons this case will have to be considered separately. CHAPTER II. SOLUTION OF THE DIFFERENTIAL EQUATION. IF we denote the operation x by ~, the differential equation d2y dy 2X2+ + (X - n2) y= 0 dx2 dx may be written in the form y-2y + (x2 - 2) y = 0. Assume that there is a solution of the form y = x (ao + ax + acx2 +...) = X,%xr.+s; 0 then, if we substitute this expression in the left-hand side of the differential equation, and observe that ~xrn = manm, the result is (r- n) taox' + {(r + 1)2 - n2} a1xr+ + E [{(r + s)2 - n2} as + as-2] Xr+s 2 The equation will be formally satisfied if the coefficient of every power of x in this expression can be made to vanish. Now there is no loss of generality in supposing that a0 is not zero, hence the first condition to be satisfied is r2- _n = 0, or r= + n. In general, neither of these values of r will make (r + 1)2 - n2 vanish*; consequently a, = 0, and all the as with odd suffixes must be zero. ' An exception occurs when n=~, r= -; but this does not require separate discussion, since in this case we still have the distinct solutions yi and Y2 with n=1. The only peculiarity iss that leadsto both of these solutions. 2~ Th nypclirt sta le 8S SOLUTION OF THE DIFFERENTIAL EQUATION. [II. If we take r = n, we have s (2n + s) a, + a_-2 = 0, a0 and hence a2 = (2 2) 2 (2n + 2)' a2 ao Ca~ 4 4(2 + 4) 2.4.((2n+2)(2nz+4)' and so on. A formal solution of the differential equation is therefore obtained by putting i (. x2 x4 _ _ Y = Y1 = aoxl (i 2 y = 0 1 - 2. + 2) + 2.(2+2) (2n+ 4) *" (_)s,x 2S 4. 2) 2.4... 2s.(2, + 2)(2,4 + 4)... (2n + 2s) In a similar way by putting r=- n we obtain the formal solution X2 X4 -/ ax7 I4 X+ y9= a0o- 1l + 2 (2n- 2) +. 4~. (2n- 2) (2-4) + " which, as might be anticipated, only differs from y, by the change of n into - n. If n is any finite real or complex quantity, except a real integer, the infinite series which occur in y1 and y2 are absolutely convergent and intelligible for all finite values of x: each series in fact ultimately behaves like X2 X4 X6 x- + x.. 22 22. 42 22. 42. 62 the rapid convergence of which is obvious. The ratio of y, to y2 is not constant; hence (with the same reservation) the general solution of the differential equation is y = Ay + By2, A and B being arbitrary constants. If n = 0, the integrals y, and y2 are identical; if i is a positive integer y2 becomes unintelligible, on account of the coefficients in the series becoming infinite. Similarly when n is a negative integer Y2 is still available, but y, is unintelligible. In each of these cases, therefore, it is necessary to discover a second integral; and since n appears only in the form of a square in the differential equation, it will be sufficient to suppose that n is zero or a positive integer. II.] SOLUTION OF THE DIFFERENTIAL EQUATION. 9 In accordance with the general theory of linear differential equations, we assume a solution y = (ao + bo log x) x-' + (a1 + b& log x) x-'+1 +.. = x-92S (a, + b, log x) xs, 0 then, observing that 32 (xm' log x) = jmxn log x + 2mx', and making a few easy reductions, we find that this form of y gives 2'y + (x2 -_ n) y =- 2nbox-w1 + {(- 2n + 1) (a1 + b1 log x) + (- 2n + 2) bj} x-1l 00 + {Is (- 2n + s) (a, + bL log x) 2 + (a,-2 + bS-_ log x) + (- 2n + 2s) bs} x-1+S. The expression y will be a formal solution of the differential equation if the coefficient of every term xg-ns or x-n+s log x on the right-hand side of this identity can be made to vanish. In order that this may be the case it will be found that the following conditions are necessary*:(i) The coefficients bo, b1, b2,... b2,_1 must all vanish. (ii) All the as with odd suffixes must vanish. (iii) The coefficient a0 is indeterminate; a2 - a 2(2n-2)' a2 ao a4 4 (2n- 4) 2.. (2n- 2)(2n -4)' and so on, up to, ___________Oo________~_0 21-' = (2. 4...(2n-o2) (2-2) (2n- 4)... (iv) The coefficient ac, is indeterminate; 7 a1 2Cn-2 blb = 2n ' 21+-2 = -22 — b21L+< —! 2 (2n- + 2) 2. 2n (2n + 2) ' bL +. 1 = (_)_^ (s > o ) ^ - ' 2.-4... 2s. 2n(2n + 2)...(2n + 2s)' ~ The reader will perhaps follow the argument more easily if he will write down a few of the first terms of the sum, and also a few in the neighbourhood of s = 2n. 10 SOLUTION OF THE DIFFERENTIAL EQUATION. [II. and all the coefficients b2?+,+, b2,+3, etc. with odd suffixes must vanish. (v) Finally a2n 2n+4 a,),,,+2 — b22n+2 + 2 (2n + 2) 2 (2n + 2) 2 a_2? 1 (1 I 1 b (2 + ( 2 + 22) 2 r2n + 2 2n Ct2l4+2 (2n + 8) b a2~+4 = -4 (2n + 4) 4 (2n + 4) _ a2?12 1 2. 4. (2n + 2) (2n + 4) 2. 4. (2n + 2) (2n, + 4) (1 1 1 1 2 4 2n+-2 +2 + "21n, and, in general, when s > 0, (-)s c2',,++2= 2.4...2s(2+ 2)(2 + 4)...(22 + 2s) {ctan - b2 ( + 2 ) I (~+ 2n+ ' All the coefficients which do not vanish may therefore be expressed in terms of two of them; if we choose a(t2 and b21, for these two, y assumes the form y = a2nyi + b2ly2, where y, = 1I - ^ _^ +....-.. 7 where yl=X~{1-2 (2tt+2)~ 2.4(2 t+ 2)(2n + 4) (the solution previously obtained), and y = xb - C'x" 2 ) lY 2 = + - l2+) 1log - X T212 + 2) 2.4.(2n- 2) 2(2n-2 +...4) 2 _,X-9?+2 X-?b+4 -2.. (2n + +2 (2n - 2) 2. 4) (2 - +2) (2n2 - 4+ x?-2( 2. 4. (2n + 2)(2+4) 2 + -4+2n+2 + 2q4J +.* + (-)S-f 9 + 2.4... 2s (2n + 2) (2n + 4)...(24 + 2s) s ( 21 + 2s) +... 8 II.] SOLUTION OF THE DIFFERENTIAL EQUATION. 11 The characteristic properties of the integral y, are that it is the sum of y, log x and a convergent series, proceeding by ascending powers of x, in which only a limited number of negative powers of x occur, and the coefficient of xM is zero. It becomes infinite, when x = 0, after the manner of x-n; for any other finite value of x it is finite and calculable, but not one-valued, on account of the logarithm which it involves. The general solution of the differential equation is y = Ay + By,, A and B being arbitrary constants. When n = 0, y2 does not involve any negative powers of x, and its value may be more simply written y = - + 22-... log x Y 2 = 22 + 22. 42 *-@ ) SO ~+-=(1 I + + 1++ - +22(1 +2) 2.42 (+2 3) 224. 9 It is found convenient, for reasons which will appear as we proceed, to take as the fundamental integrals, when n is a positive integer, not yi and y2 but the quotients of these by 21". n!. These special integrals will be denoted by J,? (x) and Wl (x), so that X2 X4 J ()= 2. n! 1- 2 2 (2n + 2) 42.. (2n + 2) (2n + 4) - o 00 8 X1b+2S - -- )_X ----S 10 o 22t-+28.!(n + )!' (where, as usual, 0! is interpreted to mean 1), and 2'1-01 (it - 2)1X-'4+2 Wn (x) = Jn (X) log x - 2'"n- (n - 1) + 2- (+i-2)! 2 n-5 (n -3)! X-in+4 X-2 ( + 2! + +2-l (n-1)! 22 (ii + 1 + X 1 + {2(s S ( )! 1 (2s 2 + 2n+s)} I The definition of J,, (x) may be extended to include the case when n is not integral by means of Gauss's function TIn, which is also denoted by P (n + 1). When it is a positive integer, Ini is the same as qn!; its general value is the limit, when Kc is infinite, of (, )=(+1. 2.... 1 n- (c, (it) = (I + 1) (}a + 2)...(it + to) t' 2 12 SOLUTION OF THE DIFFERENTIAL EQUATION. [II. The function In is intelligible and finite for all real finite values of n, except when n is a negative integer, when Hn becomes infinite by a convention we define I0 to be 1. It will be well to recall the properties of IIn which are expressed by the formulae HIn = nr (n - 1) I (-n) n ( - 1) = 7 cosec n 13 I (1) = 1V7r (Cf. Forsyth, Treatise on Differential Equations (1885) p. 196; Gauss, Werke III. p. 144.) The function Hnt may also be interpreted when n is complex; but we shall not require this extension of its meaning. This being premised, the general definition of Jn (x) is given by the relation O H S X(-) W+28 Jn (x) = o 2l+2s s Al (n +); I4 and in like manner, if n is not a positive integer, oo, (_)S X-)6+2S J_, (x) = _,+2 Hs (-+) * 14 0 ITIS H (- n + s) ' If n is a positive integer, the function J_, (x), properly speaking, does not exist; it is, however, convenient in this case to adopt the convention* expressed by the formula J-n (x) = (-)1 J(x (x), [n integral]. I 5 When the argument x remains the same throughout we may write J,, instead of J, (x), and indicate differentiation with respect to x by accents; thus J$, will mean J (x), and so on. dx By differentiating the general expression for J, (x) we find that _ (-)(n + 2s) 11+_-i 0 2~b2 Hs IT (ist + s) i t X (-)HS Xl+25-1 JI n (l)s Xlr2s++l x 0 22S Ins I (n + s + 1) " It may be proved that if J_,_ is defined as in (14') the limiting value of J(,,+e) when e is infinitesimal, and n a positive integer, is precisely (- )"J,: so that the convention is necessary in order to secure the continuity of J,. II] SOLUTION OF THE DIFFERENTIAL EQUATION. 13 (on writing s + 1 for s); that is, JXzI-J - J1+1; I6 or, which is the same thing, Jn+1 = r n J- Jn I6' Again, writing n + 2s in the form (n + s) + s, we have (_)S x+2-1 x (_)S 1+2S-1 + J= 0 22+2S Hs H (t n+ s-1) 1 2+28 H (s -) (n + s) = (Jn- - Jn+l). 17 It will be found that with the convention 15 these formulae are true for all values of n. It is worth while to notice the special result J =-J. I8 If J,,+ is eliminated by combining I6 and I7, the formula J1Jn-^-^Jn 19 is obtained; and similarly by eliminating J' it will be found that 2n J,1 - - Jnr + Jn+1 = o. 20 The formule 16-20 are very important, and are continually required in applications. It follows from 17 that 4J = 2J', - 2J1 = (Jn-2 - Jl)- (J,,- J-2+2) = J-2 - 2Jn + Jn+2, and it may be proved by induction that 2sJ') = J - SJS+2 + S J-(s -) - + (_)S Jn+, the coefficients being those of the binomial theorem for the exponent s' 14 SOLUTION OF THE DIFFERENTIAL EQUATION. [II. The analogous formule for WT,, which may be obtained by a precisely similar, although more tedious, process, are t Jin+l x 2 (n ~-1)' it Jot- l WI-, = WI,- - - nn + 22 X 2n J Wn+l - We+ Wit-l ~ 1+ ~Vr — 23 l,1-x 2 (n +1) 2n3 n l "1 Now if we put 2-n = -, 24 1 s Yn, = W - anJd, 25 with the convention 00 = 0, Y, is a solution of Bessel's equation which is distinct from J,, and which moreover satisfies the relations _ya =_ - Y- y, 26 x Y'= Yn-1 -Y- Y, 27 x 2n Yn+ — Y, + Y_- = 0, 28 x Yo=- Y, 29 which are of exactly the same form as 16, I9, 20, I8. The explicit form of Y, is Yo = J log {- 2'-l (n - 1)! x-n + 23 (n-2)! X-n+2 2n-, (n - 3)! x-+ + 2-2 + 2! (-) +211-1( 1)! Xn En 1 (_)S-] k1al S 1n+2s wh2n1ere 2 (S +1 3 1 The n w d b. N ann, and my I 2s 2 (i + s) 2s The function Yn was discovered by C. Neumann, and may be referred to as the Neumann function of the nth order. The explicit form 30 was given by Schlafli (Math. Ann. III. (1871), p. 143), who obtained it by a different process from that here employed. II.] SOLUTION OF THE DIFFERENTIAL EQUATION. 15 There is another way of deducing a second solution of the differential equation which deserves notice. In that equation put y= uJ,(x), where u is a new dependent variable: then, observing that J,(x) is an integral, we find that d2jt (1 du Jn (x) d2 + J (x) + 2J,; (x)) -d = 0; d2u or d" 1+ 2- 4 ( — 0 du x Jn (x) dx du whence xJ (x) d = A, dx u=AJ xJ() B; and y AJ (x)J (x+ BJr (x) 32 is the complete solution of the equation. By properly choosing the constants A and B it must be possible to make y identical with J_, (x) or Y, (x) according as n is not or is an integer. Taking the general value of Jn (x), n not being an integer, XJ X- (x) = " (n)221+1 +*, fx dx _ 221 -so that -td = -- 222-ln (n -1) x-2 + and A Jn(x) - 2- 1 (n - 1)Ax-n +.... ' JlxJ (X ) v 2n The leading term of J-n (x) is - (n x-'n, and by making this agree with the preceding, we obtain 2 2 A =-n (-)=- sln zr. 33 l (n - 1) H (- n,) The value of B will depend upon the lower limit of the integral [x dx f J,' (X) 16 SOLUTION OF THE DIFFERENTIAL EQUATION. [II. Divide both sides of 32 by J, (x) and differentiate with respect to x: then in the case when y = J_, (x) we have d {J-n) 2 sin nwr dx J J ) rXJ, ~~/ 2 or J/J-n_ - J._'JI = -- sin n7r. 34 7tX With the help of I6 and 19 this may be reduced to the form 2 JnJ-n+l + J-nJn-i = sin n7r. 3 5 7rX This suggests a similar formula involving Neumann functions, when n is an integer. If we write Ut = X (Jn+ Y"n - JqnYn+li) we find with the help of I6, 19, 26, 27 that du d = Jn+1 Yn - JI Yn+ + Yn (xYJ- n + 1Ji+1) dx n + J1+1 ( Y1 - xYq+1) - J (XY? - n + 1 Y,+,) - Yn+1 (nJ- xJ,,+,) =0 identically. Hence i is independent of x: and by making use of the explicit forms of J, and Y, it is easily found that the value of u is unity. We are thus led to the curious result that J,+13 Y1n- J,~ n+ = -. 36 x Returning to 32, we find, by comparison with 30, that, when n is a positive integer, fX dx Yn = -xJ + BJn. 37 As in the other case, the value of B depends upon the lower limit of the integral. CHAPTER III. FUNCTIONS OF INTEGRAL ORDER. EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. THROUGHOUT this chapter it will be supposed, unless the contrary is expressed, that the parameter n, which occurs in the definition of the Bessel functions, is a positive integer. The expression for the Bessel function of the nth order is T /(\- (-)sXn+2 oJ 2nf+Ssi (n + s) which may be written in the form o0 Xn+s (_ X)8 o 2n+8.(nz+s)! 28.s! hence we conclude that Jn (x) is the coefficient of t' in the expansion of exp (t- according to powers of t. In fact, X xt - xt-1 ex (t t-1) exp exp exp E XItr X (_)SXst-s o 2r.r!' 2S.s! (._)sXr+s t-s 2r+sr! s! and the coefficient of tn, obtained by putting r - s = n, is (_)sxn+r2s _ = oL (X). 216+28 (n + s)! s n In the same way the coefficient of t-" in the expansion is (-) Jn (x), or Jn (x), so that we have identically exp 2 (t - t-') = VJ- (x) tn. 38 expG 38 - 0 G. AM. 2 18 FUNCTIONS OF INTEGRAL ORDER. [III. The absolute value of Jn+ (x)/Jn (x) decreases without limit when n becomes infinite: hence the series on the right hand is absolutely convergent for all finite values of x and t. Suppose that t = eit; then the identity becomes eixsinl = Jo (x) + 2iJl (x) sin # + 2J2 (x) cos 20 + 2iJ3 (x) sin 30 + 2J4 (x) cos 40 +... 39 and hence cos (x sin b)= Jo (x) + 2J2 (x) cos 2 +~ 24J (x) cos 4b +... 40 sin (x sin b) = 2J1 (x) sin b + 2J3 (x) sin 3b + 2J5 (x) sin 5~ +... 41 Change b into 7 - b; thus cos (X cos ) = Jo (x) - 2J2 (x) cos 20 + 2J4 (x) cos 40 -... 42 sin (x cos b) = 2J, (x) cos - 2J3 (x) cos 30 + 2J5 (x) cos 50 -... 43 These formulae are true for all finite values of b. Multiplying the first of these four formulae by cos nb0 and integrating from 0 to 7r, we obtain I7r cos -n cos (x sin 0) do = 7rJn (x), if n is even (or zero), o =0, if n is odd, or, in a single formula, fnr 7r cos n cos (x sin 0) d0: = 2 {1 + (- 1)} J, (x). 44 Similarly sin nbi sin (x sin 0) dob = {1 - (). 45 By addition, f cos (nb - x sin o) do = 7rJn (x), 46 which holds good for all positive integral values of n. It will be remembered that an integral of this form presented itself in connexion with Bessel's astronomical problem; in fact we arrived at the result that if, = 0 - e sin b, 00 then 0 = b+ E A, sin rp, where Ar cos r ( e sin ) where Ar- cos r (0 - e sin 0) do. rrrJ. EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 19 We now see that Ar may be written in the form 2 Ar - Jr (re), and in this notation =, + 2 {1 (e) sin / + J2 (2e) sin 2 + J3 (3e) sin 3/ +.... I 2 3 J It is known that 772 nos =- ('T - ) ( 3) 2 cos n = (2 cs) (2os C (2 c + ( cs -4 s! 22! +(-)n (n - s - 1)( - s - 2)...(n - 2s+)(2coso-2+ 8 i Now let this transformation be applied to the identities 42 and 43, and let the expressions on the left hand be expanded according to powers of x cos At: then by equating the coefficients of cosn f on both sides we find xZ = 2n. n! Jn + (n + 2) Jn+2+ (n ) Jn+4 2! (n + 6) (n + 2) (n + 1) + 3! 1 (n + 2s) (n + s - 1) (+s - 2)... (n. + l}) 4 + 8 X- J-+2S + *** 7 47 which holds good for all positive integral values of n, and also for n = 0. The first three cases are 1 = J+ 2J2+ 2J4+... + 2J+... x = 2J1 + 6J3 + 10J5 +... + 2 (2s + 1) J2,+ +. C2 = 2 2(4J + 16J4 + 36J +... + 4s2Js +...). In confirmation of these results it should be observed that the series in brackets in 47 is absolutely convergent; for if we write it `ics Jn+2s we have 1- x... Cs+, JI+2.+2 _ (n + s) x2 2 (2n4s + 6) csJn+2~ 4(s+l1)(n+2s)(n+2s+l) l x2 2(2n+4s+2) and this decreases without limit when s increases indefinitely. 2-2 20 FUNCTIONS OF INTEGRAL ORDER. [III. Moreover if Sh denotes the sum Co Jn + c1 Jn+2 + *. + Ch Jn+2h, it can be proved by induction that = o - + E )r~ (n + h)l(ri - 1) (r - 2)..(.. - /) 2n.'z:- Sh (+h(-)+ -h ' —1 2"2r (n+ r +h)! rh! h x or, which is the same thing, On. n n!Sh = Xn V7(n + h)! xl~+2h+2 22h+2 (n+ 2h +1)! (h + 1)! {^ X2 )... j2 22(h +2)(n + 2h +2). 2!+ Since the series in brackets is convergent, the expression on the right hand may be made as near to xn as we please by taking h large enough; moreover the series C Jn+2s s=h+l becomes ultimately infinitesimal when h increases indefinitely; therefore the relation 00 x 2 = 2nn! Y cs Jn+28 0 is true and arithmetically intelligible for all finite values of x. Now suppose we have an infinite series axs = ao - + ax + a2,2 +..., 0 then on substituting for each power of x its expression in Bessel functions, and rearranging the terms, we obtain the expression 00 z bsJs = boJo + bJ+ + b2J, +-, 0 where b= ao, b = 2a1, b2 8a, + 2ao,... and, in general, bs, + 2(9.s)! ia,+ (s- 3) 1 a 4 22 *S - )' I! 24 ( 1)0?2)' 2! + a,,, -,+ 48 26(s ) (s-2)(s-3) 3! * 4 the sum within brackets ending with a term in a. or ao according as s is odd or even. III.] EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. If the series Sax8 and XbJ, are both absolutely convergent, we may put x2aS8 = 2b8Js, 49 and the arithmetical truth of this relation may be verified by a method similar to that employed above. An important special case is when the series csas satisfies Cauchy's first test of convergence; that is to say, when for all positive integral values of s above a certain limit, a,x <asx Ias-, where K is a definitely assigned proper fraction. In this case the limit of bsJ/b,_sJs,- is ax/as_1, and the series Eb8J, is absolutely convergent. Cases in which ax/as_i is ultimately equal to unity have to be examined separately. If we are assured of the possibility of an expansion such as that here considered, the coefficients may be found by any method which proves convenient. As an illustration of this, let us assume Y0 = J0 log x + ZCn J~. Then Y = J0 log x +Jo + cJ', 29 Jo Yo = Jo' log x + - + _ cJ; and hence 0 = Yo'' + Y' + Yo = Jo' +S (Jc J+ 1 J -+ Jn) 2J n2n X-2 Therefore nEn2cJn = - 2xJ' = 2xJ1, by i8. Now by repeated application of 20 xJ, = 4J2- xJ3 = 4J - 8J4 + xJ5... =2 (2J2- 4 + 6J -...) 22 FUNCTIONS OF INTEGRAL ORDER. [III. Hence- n2CCJn = 4 (2J2 - 4,4 - 6J6 -), and finally Yo= Jolog +4( J2-J4+ J6-)- 50 Differentiate both sides with regard to x, and apply I7, I8, 29; thus Y1 = J, log x - + 4 J3 J) (JJ - J- (J3) +. 6.. that is, Y=& o log- I o J+43-+ - I Yx 2.4 4J3 6 J5 t. -6. 87 5 Now we have identically, by 40 or 47, 1 = J+ 2J2+ 2 + +..., r 1 1 2 2 and therefore - --- - J2 + J4+... w x x x 11 1 __+ (J1+J3)+ (J3&+J)+ -=+ 1 ~ J3) - 3 J +..4 1 2 2.5. 6 Consequently I 1 6.3 2.5 6.7 Y,=J logx- - - J+ J3-J - J5+ - J... x 2 2.4 4.6 6.8 6 (4s + 3) J 2 (4s + 5) -t( s + ( ) (4s+6) +~ 4S+'" 52 (4-s + 2) (4s + 4) + (4S + 4) (4s + 6) Similarly 2J; 2J1 3 Y,= J2log- x2 x 2J2 j}4 4J4 66 68 s3 } (2.6 4.8 6.10-' ' which may be transformed into Y, = J2 log - -L 2- - 4 J (1.3i 2.4f)9 (3.j 4. 6) + 13.7 J14 + 54 5.87 +'"' III.] EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 23 or again into Y n ^2 1\ 3 T2.4j 1+.3 -=J2log x - (+2 - J + J- J + 2 6 +... (2 2 1.3 2s (2s + 2) (2s- 1) (2s + I) (2s - 1) (2s + 1) 2s (2s + 2) J+ + ' 55 Neumann has given a general formula for Y, which may be written in the form Yn,=Jn1ogx-l ++i + 4... + Jn Y"' = Jiagr-iI _Jn l2 3 n/!~ -1 1 2 n-8 Js n + 2s n'- (1 s ( + Jn+os. 56 2 fl-s Si 1 s (n + 8) This may be proved by induction with the help of 20 and 28. There is another interesting way of expressing Yn. If we write 2n-3 (n - 2)! T = 2'-1 ( - 1)! x- + 1!2 +2 2n-5 (n - 3)!!_-2 )-'+4 + '" + 2n-1 (! 57 a polynomial which has already appeared in the expressions for Wn and Yn, then we have W= Jn log x - Tn + 2 ( + {n +2} Jn+2 2 (n+ 1 n +4 (n + 2) {2! - n ) 2} J+4 n+6 n(n + ) (n + 2) } + 6( + 3) 3! 4+2J~... n+2s {n(n + 1)... ( l + s - 1) - )S } 2s (n+ +s) s! + + -...'12 tsJ-1+) 2 +38..., 58 and the corresponding expression for 1can at oe be wrin can at once be written down. Other illustrations of these expansions will be found in the examples at the end of the book. 24 FUNCTIONS OF INTEGRAL ORDER. [IlI. A great number of valuable and interesting results are connected with a proposition which may be called the addition theorem for Bessel functions; some of these will now be given. By 38 we have J (u + v). tn = exp - (t - It t 1\ v / 1\ exp (t - t)exp 2 ( t) t-2 t 2 -t = XJn (u) tn C Jn (v) tn. - GO -CjO Multiply out, and equate the coefficients of the powers of t on both sides: thus Jo (u + v) = Jo (It) JO (v)- 2J1 (u) J1 (v + 2J ( v) J () -... J1 (U + ) = Jo () J (v) + J1 (t) Jo (v) - J1 (t) J2 () - J2 (t)) J1 (v) +..., and in general n Jn (it + V) = Jgs (n1) Jn-s (v) 0 + z (-)8 {J (J ) Jn+s (v) + Jn+s (1) Js (v). 59 1 Observing that J, (- v) = (-)J(v), we find that if n is odd 2 {Jn (u + V) + Jn (It - V)} = J1 ('1) Jn-1 (v) + J3 (I) Jn-3 (V) +... + Jn () JO (v) - J1 (it) JSn+ (v) - J3 () Jn+3 (v)-. + Jn+2 (U) J2 (V) + Jn+4 (u) J4 () +..., 60 and if n is even 2 tJn (U + V) + Jn ( - V)} = Jo () Jn (v) + J2 (u) Jn-2 (v) +.+.. + Jn (u v) + J2 (U) Jn+2 (v) + J4 (U) Jn+4 (v) +.. + Jn+2 (u) J2 (v) + Jnl+4 (u) J4 (v) +.... 60' III.] EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 25 Similarly if n is odd I Jn (U + v)- Jn (U - v) = Jo (u) Jn (v) + J2 (U) Jn-2 () +... + Jn-1 (u) J1 () +- J (u) Jn+2 (V) + J4 (u) Jn+4 (v) +... - J+1 () J1 (v) - Jn+3 (u) J3 (v) -.. 6 and if n is even 1 {Jn (U + ) - Jn ( - v)} = Ji (u) Jn-i (v) + Js () Jn-s (V) +.. + Jn-1 (u) J1 (v) - J1 (t) Jn+1 (v) - J3 (u) Jn+3 () -... - Jn+ () Ji (V) - Jn+3 (Wu) J3 (V) -.... 6lI By putting u = x and v = yi, where x and y are real quantities, we obtain from these formule expressions for the real and imaginary parts of Jn (x + yi). It should be noticed that Jn (yi) immediately presents itself as a real or purely imaginary quantity according as n is even or odd. We will now consider a remarkable extension of the addition theorem which is due to Neumann*. By 38 we have exp 2( kt- = 1 knJn (x) t; x, 1\ 1\ x, 1 now (t- ) =k2(t-) +(- k) k, therefore exp (kt - =t ( exp\ xp (. t- + 00c /J\ that is,, knJn (x) tn = e\l +2 Jn (k1x) tn. 62 "~~ -00. 62 Put x = r, k = e0i: then +oo ir sin 0 +oo 2 Jn (re0i) tn = e t enJn (r) t. 63 -00 -00 * Strictly speaking, Neumann only considers the case when n =0; but the generalisation immediately suggests itself. 26 FUNCTIONS OF INTEGRAL ORDER. [III. Equating the coefficients of tn, we have ir sin 0 Jn (reei) = eniJ (r) - 1 e (n+l) OiJ (r) iz2 sin2 + -! (n e2)in+2 (r)-... 2! = n + ni, 64 where t = Jn (r) cos nO + r sin 0 sin (n + 1) OJ,n+ (r) r2 sin2 0 - -2 cos (n + 2) 0Jn3(r) r2 sine 0 - r s 0 sin (n + 3) J+3 (r)+..., 64' 8! anid 7,W = Jn (r) sin nO - r sin 0 cos (n + 1) 0 Jn+ (r) r2 sin2 0 2-,i! sin (n + 2) J,+2 (r) 2 I + r i- 0 cos (n + 3) 0~Jn+3 (r) +.... 64" As a special case, let =; thus r2 3r ) Jn (ri) = i {j, (r) + rJ+, (r) + 2 Jn+2 (r) + j Jn+3(r) +.... 65 Returning to 63 let us put r, 0 successively equal to b, /3 and c, y and multiply the results together; thus +00 +00 I Jn (bei) tn" Jn (ce>i) tnl -00 -00 i(bsin S+csiny) +oo +00 = e t eniJn (b) tn E e"YiJ, (c) t. 66 -00 -00 Now the left-hand expression is equal to - 00 and if the right-hand side is expanded according to powers of t, the coefficient of t, gives an expression for Jn (beP + ceYi) in the form Jn (bei + ceVi) = C - C i (b sin /3 + c sin y) C2z (b sin,8 + c sin ry)2 + -21..., 67 2! III.] EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 27 where Co = eniJ (b) Jo (c) + e{(n-1) +Y}i Jn_(b) J,(c) +... + enyiJo (b)Jn(c) - e{(n+1) P-v}Jn+1 (b) J1 (c) + e {(n+2) P-2y}iJ+2 (b) J2 (c) -.. - e{(n1+) r-}i J1 (b) Jn+l (c) + e{ (+2) y-2}iJ (b)Jno, (c) -.., 68 and in like manner for C1, C2, etc. Since, however, this formula is too complicated for practical purposes, we shall only consider in detail the case when bedi + ceei is a real quantity. Moreover we shall suppose in the first instance that n = O. If we put befi + ceei = a, a real quantity, we have a2 = (b cos / + c cos 7)2 + (b sin / + c sin 7)2 = b2+ 2b cos ( - y7)+c2, and also b sin / + c sin y = 0. Let us put / -r = a; then the general formula 67 becomes in this special case Jo (/b2 + 2bc cos a + c2) = Jo (b) Jo (c) - 2J1 (b) J1 (c) cos a + 2Ja b) J2 (c) cos 2a -... =Jo (b) J (c) + 2 (-)Js(b) J,(c) cos sa. 69 If we change a into 7r - a, the formula becomes Jo (,^/b - 2bc cos a + c2) = JO (b) Jo (c) + 2> JS (b) J, (c) cos sa, 69' and this is Neumann's result already referred to. By way of verification, put a = 0; then we are brought back to the addition formulae. 77 Suppose a= 2; then we have Jo (,/b + c2) = Jo (b) Jo (c)- 2J2(b) J,(c) + 2J4(b) J(c) -... 70 and hence, by supposing c = b, J (b 2) = J2(b) - 2J2(b) + 2J4(b) -.... 28 FUNCTIONS OF INTEGRAL ORDER. [III. In 69 and 69' suppose b = c; thus J0 (2b cos = J(b) - 2J, (b) cos a + 2J2 (b) cos 2a -. 2 7 72 Jo (2b sin = J(b) + 2J: (b) cos a + 2J2 (b) cos 2a +. In order to obtain another special case of 67 let us suppose that n = 1, still retaining the condition b sin 3 + c sin y = 0. Since the four quantities b, c, 3, y are connected by this single relation, there are three independent quantities at our disposal. Let us choose b, c and (/ - y), for which, as before, we will write a. Then if a has the meaning already assigned to it, it may be verified, geometrically or otherwise, that aesi = b + ceai, aey! = c + be-a. Now when n = 1, the formula 67 gives aJ1 (a) = aesi {JJ (b) Jo (c) + e-i Jo (b) J, (c) -eai J2 (b) J,(c) e2ai J3 (b) J2(c)-... -e-iJli (b) J, (c) + e- a J (b) J, (c) -... Substitute b + ce'a for aeti on the right-hand side, multiply out, and equate the real part of the result to aJ, (a); thus aJl (a) = {bJ, (b) Jo (c) 4- cJo (b) J1 (c)} - {bJ2 (b) J1 (c) - bJo(b) J, (c) - cJ, (b) Jo (c) + cJ, (b) J2 (c)} cos a + tbJ3 (b) J, (c) - bJ,(b) J, (c) - cJ (b) J, (c) + cJ, (b) J3 (c)} cos 2a = bJ1 (b) Jo (c) + cJo (b) J J(c) + X (-_) [b [Jn+l (b)- J,_ (b)] J, (c) + c [Jn+l (c) - Jn-_ (c)] Jn (b)} cos na. 73 This result may also be obtained from 69 by applying to both sides the operation a a 8 = b a+C and in fact this operation affords the easiest means of obtaining the formulae for n = 2,:3, 4, etc. ITI.] EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 29 Special results may be obtained from 73 like those deduced from 69. For example, putting b =c, we find, after a little reduction, that J1 (2b cos 2 = 2Jo (b) J1 (b) cos -2J, (b) J, (b) cos 32 5a + 2J, (b) J3(b) cos.... 74 This may also be deduced from 72 by differentiating with respect to b. In 72 and 74 write x for b, expand both sides according to powers of cos, and equate coefficients; thus 1 = Jo2 () + 22 (x) + 2J22 (x) -.. -2 = 4 {J2 (x) + 4J22 () + 9J3 (x) +...}, and in general ^-+2'-' J~~1+ 2(~ + 2 2)(2+ l)?=2n = 2~n 2 (!)~ - J2 + 2 (n + 2) (2n + 1) + \ t2'a I r1 2! ' " +2 (n + s)(2n +1)(2n +2)...(2n s- 1)J2 + 75 while for the odd powers of x X= 2JoJ + 6JiJ2 + 10J2J3 +... + 2 (2s + 1) JJ8+ +.. X3= 16 {IJJ+ 5J2J3 + 14J3J4 +... s (s + 1)(2s+ 1) js. + 6 ss+ + ' and in general 2n- = -22n-1 n! (n-1)! {JnJn + (2n + 1) JnJn — + n (2n + 3) JTn+1%J+2~ 2n (2n + 5) (2n + 1) + + + n (2?n + 3).nl,+&,+ + 3- nn+3 +... 2n (2n + 2s- 1) (2n + 1) (2n + 2)... (2n + s- 2) J +.... 76 30 FUNCTIONS OF INTEGRAL ORDER. [III. 0 By means of 75 and 76 the series faSx8 may be transformed into boJ + byJoJl + b2J1 + bJ2 +... when bo = ao, b. = 2ai, b2= 4a2 + 2ao, b3 = 16a3 + 6a,, b4 = 64a4 + 16a2 + 2ao, and so on. The arguments employed already in a similar case will suffice to prove that we may write 00 00 2axS = S (b2SJ + 2Sb,+2 sJ+i), 77 0 0 and that this is arithmetically intelligible so long as x remains inside the circle for which Saxs is absolutely convergent. This result is also due to Neumann (Leipzig Berichte 1869). We will conclude this chapter by a proof of Schlomilch's theorem, that under certain conditions, which will have to be examined, any function f(x) can be expanded in the form 1 f(x) = -a, + aiJ (x) + a2Jo (2x) +... + aJo (nx) +..., 78 where{ / Jr < dt6, where ao =f 2 f(0) + o j} di, and, if n > 0, 2 J7r {ff' (O^) d 7 a,, = - U Cos Mn 1 - = da. To prove this, we shall require the lemma 7r r2. \ -cos?m J (nu sin n) d = 1- cos 79 The lemma may be established as follows. We have J,(nt sin ) = * (-) n2S+l U2s+l sin2s+l. o 22'+si (s + 1)! therefore J, (n sin ) do - 2 28 sin ) +i+i J' 1 2+l!( 1)! o 2 fi2i - (-_) n 2S+1 t2S+l 2S1 s! = 8! (- + 1+ o 2"+'s!(s +)!' 1.3. 5...(2s + 1) - (_)S n2S+1 U2S+ 1 _ cos nu o (2s+ 2)! nu EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 31 Now if we assume f(x) = - ao + aJo (x) +... + aJo (nx) +... we shall have f' (x) = - alJ (x) - 2a2J1 (2x) -... - nan (nx) -.... Write u sin f for x, and integrate both sides with regard to q between the limits 0 and 2; thus 2i vr jf / (u sin c) do - nan a J1 (nu sin b) do _ an (cos,- 1); U and therefore 7r u f' (u sin b) dCb = an cos nu - an 0 1 1. Cancos nu + aO f(O) s. Hence 7r ao -(0) =- J {a (t sin ) d d or, which is the same thing, a= - {(0) + / ( sin b) d du, and, when n > 0, 7r 2 7' 2 - it mcosnu {j f s (u sin ) ) d0bj du. On putting sin k = f, we obtain the coefficients in the form given by Schlomilch; but it must be observed that the theorem has not yet been proved. All that has been effected is the determination of the coefficients, assuming the possibility of the expansion 78 and also assuming that the result of differentiating the expansion term by term isf' (x). 32 FUNCTIONS OF INTEGRAL ORDER. [III. In order to verify the result a posteriori, let the coefficients ao, a1, a2, etc. have the values above assigned to them, and let us write, (x) = a2 o + aJo (x) + aJo (2x) +.. = f(0)+ + Jo (x) cos u + Jo (2x) COS 2u +...) F(u) du, 'r 2 r2 where F(u) = f ' (u sin k) do). 7r We have 7 Jo (nx) = I cos (nx sin )) do 2 Jo - fC cos n d /J o 32 _ u2 on putting x sin c = u. Now if x lies between 0 and 7r, we are entitled to assume an expansion 0 (u) = o + CU cos C + 2 + cos 2u +..., 8 (u) being equal to (x2 u_2)-f from u = 0 to u = x, and equal to zero from u = x to u =7r; the isolated values u = 0,, r being left out of consideration. By the usual process, 7rco= 0 (u) du = - 2 s Jo0 =o Vx,2 - u2 therefore Co= 2 while if n > 0 7 FT cxirr 7 fos nu du - Cn =J (u) cos nu du = VX --- 2 J\ Jo /X2_U 7r = Jo (nx). Consequently cn = Jo (nx), and 0 (u) = + J (x) cos u+ J (2x) cos 2u+... 2t EXPANSIONS IN SERIES OF BESSEL FUNCTIONS. 33 This is the series which appears in r (x), and since the value of 0 (a) is known for the whole range from 0 to wr with the exception of an isolated discontinuity when t = x, we have + () =f(0) + F() 7rJo V '2,2 7r =f(O) + f2 X f'(u sin 4) tudadub Put i sin u = A, u cos qb = D; then this becomes 2 s X2 r- f '(ds ae, (x) = f(0) +2 ) d=f (0) + {f() r f ()} =f (). This shows that Schlomilch's expansion is valid, provided that x lies between 0 and 7r, and that the double integral f X' (a), dra J J 72 _ a2 _ 2 taken over the quadrant of a circle bounded by:=0, 7 =0,:2+k 2- X2 =0 admits of reduction to { f(x) -f(0)} which is certainly the case, for instance, if f(x) is finite and continuous over the whole quadrant. G. M. 3 CHAPTER IV. SEMICONVERGENT EXPANSIONS. THE power-series which have been obtained for Jn and In -Jon log are convergent for all finite values of x, but they become practically useless for numerical calculation when the modulus of x is even moderately large; it is therefore desirable to find expressions which approximate to the true values of J,, and Y55 when x is large, and which admit of easy computation. The expressions which we shall actually obtain are of the form /2 Jn = A/.- {P sin x + Q cos x} T Yn = / {R sin x + S cos x}, where P, Q, R, S are series, in general infinite, proceeding according to descending powers of x. It will appear that these series are ultimately divergent, and the sense in which the equations just written are to be understood is that by taking a suitable number of terms of the expressions on the right we obtain, when x is large, the approximate numerical values of J,, and Yn. The approximate value of JO when x is large was discussed by Poisson*, but not in a very detailed or satisfactory manner; in this chapter we shall follow the method of Stokest, which is important as being capable of application to a large number of functions of a kind which frequently occurs in physical investigations. In a later chapter the question will be discussed in a different manner, depending on the theory of a complex variable. * Sur la distribution de la chaleur dans les corps solides. Journ. de l'Ecole Polyt. cah. 19 (1823), p. 349. t On the numerical calculation of a class of definite integrals and infinite series (Camb. Phil. Trans. ix. (1856; read March 11, 1850), p. 166; or Collected Papers, n. p. 329). On the effect of the internal friction of fluids on the motion of pendulums (Camb. Phil. Trans. ix. (1856; read Dec. 9, 1850), p. [81). SEMICONVERGENT EXPANSIONS. In Bessel's differential equation put J, (x)= ux-~; then it will be found that u is a solution of d2u __ __d2~ (1 - 2 = O. 8 Now when x is large compared with n, the value of (n2- )/x2 is small; and if, after the analogy of the process employed in the expansion of an implicit function defined by an algebraical equation f(u, x)= 0, we omit the term (n2 - I) u/x2 in the differential equation, we obtain d2u + u = 0, dX2 of which the complete solution is u = ut = A sin x + B cos x, where A, B are constants. We are justified a posteriori in regarding this as an approximate solution of 8I because this value of u does in fact make (n2 - ) /x2 small in comparison both with u and d2u with d dX2 Let us now try to obtain a closer approximation by putting Iu = 2-= (AO + ) sin x (Bo + 1) cos x, where Ao, A,, B0, B1 are constants. This value of u gives _12B, - (n2 -j)U (1AO j)^ d~u n- _( ) +d (1- r 4) q 1= 402Bl-(i__ -} sinx {2A1 X ) B + 2 ) B } cos x. [2 X3 ) The expression on the right becomes comparable with x-3, if we assume 2A, = - (n2-)B,, 2B1 = (z2 - ) A. The value of u2 thus becomes = Ao si 2 + - cos +Bo cosx sin 2x 2x and we have d2 2+ (1 = - - Ao cos x + Bo sin x). dx2 + 1 2x3 It will be observed that we thus obtain not only an approximate solution when x is large, but an exact solution when n = + or + 3. We shall return to this second point presently. 3-2 36 SEMICONVERGENT EXPANSIONS. [IV. Let us now assume A2 sin x + B2 cos x It = + - -; then it will be found that + _{(-~_) (n —) Bo+4Bsinx - (n 4 5) (A2 sin x + B2 cos x). If, then, we put A2~ =(-71) (- Ao, 2.4 22 2 B = _ (n-I)(n-) B 2.4 the value of u becomes u=u 3=AO sin x 4+ 2 cosa- (.4 -) sin x ( 2x 2.4 X,(12" p f) (2( 2 - X) (! 2 9 + Bo cos x- 2 sin 2 x - ( — — ) cos X and we have d3 + (1 j ) 3 = (n ( 4) (n 4) (A sin x + B cos x). Proceeding in this way, we find by induction that, if we put ur = AoUr + BoVr 82 where Ur = sin x + -- cos - -(n-(n- sin - ( (2_ 1) (2-_.(2 - (r-_ _)2). r-l + 2.6..(2r -2)-1 s 2 y 3 V, = cos x - sn (-4 cos x +... 2x 2.4x2 + (n2 -) ( -)...(n -(r -)2COS) ( r-1 84 2. 4.6...(2r —2)x - os + r, 84 SEMICONVERGENT EXPANSIONS. 37 then U2r + 1- ) Ur *"z1dX2 ^2 - u - ( W-)(2 4).. (n2- (r )) Ao sin -+ -2 r) + Bcos o + r2 7r 85 The expression ur is therefore an approximate solution so long as ( 42, _ 4) ( ). _ )(n-(r - ) 2.4.6...(2r-2) xrl is small; and tr, is a closer approximation than ur_- so long as n2 2 -. ) is numerically less than 2 (r - 1) x. It is to be observed that the closeness of the approximation has been estimated throughout by the numerical smallness of the expression dx2, 2 12,, the value of which is zero for an exact solution; that it is not to be inferred that the expression above given for Ur reflects in any adequate way the finctional properties of an exact solution; and finally, that as r is taken larger and larger the expression u,. deviates more and more (in every sense) from a proper solution. These considerations, however, do not debar us from employing the expressions Ur up to a certain point for approximate numerical calculation; and the analysis we have employed shows the exact sense in which u,. is the approximate value of a solution, and the degree of the approximation obtained. It is convenient to alter the notation by putting Ao sin x + Bo cos x = C cos (a - x), A0 cos x- Bo sin x = C sin (a - x), 86 C and a being new constants: this is legitimate, because C2=A2+B2, which is independent of x. Then we have,. = C {Pr cos (a - x) + Q, sin (a - x)}, 87 38 SEMICONVERGENT EXPANSIONS. [IV. where (4n2_ 1) (4 - 9) 1.2 (8x)2 (4n-2 1) (4n'2 - 9) (4n2- 25)(42 - 49) 88 +r 1.2.3.4(s8)4 "' Q 42n- 1 (4n2 -1) (4n2 - 9) (4n2 - 25) Q11 = -f.+ 89 v 8^~x -- 1.2.3(8 (x)3 and P,., Q, between them contain r terms, involving, x~, x-1, x... x-r'+ respectively. What remains to be done is to determine the constants C and a so that u,,x-~ may be an approximation to the special solutions of Bessel's equation which are denoted by J,, and J_n or, when n is integral, by J, and Y,. For this purpose we shall require a new expression for J,, in the form of a definite integral. Supposing that n + I is real and positive, it may be proved that J, (w) = 1 - (n ) cos ( cos ) sin2" q d 9. 90 Vw ' 2n H (It ) Jo To show this, we observe that oFTT o,,, d-E (_ )S 2S i p-) n (eS- ) o I (2s)H (n + s) Now since s is a positive integer (s _ 21) (2s-1) (2s - 3 )... 12 (-)=(2- 1.-)2"-1 ), and n (2s) = 2s I (s). (2s-1) (2s- 3)...1; therefore (s-) (-) II (2s) 22" IIs 22S Is' and Jcos (x cos ) sin 2 c, = rIn (t - ) - 2S n (s + o 2 HsT (n- + s) = 22 V/rI (n (- ). -n Jn) (w), by I4. This proves the proposition. Now consider the integral 7r U = cos (x cos 0) sin2^ 2 df = 2o cos cos ) sin 2n do, and putJoos = 1 - and put cos 5 = 1 - /J; IV.] SEMICONVERGENT EXPANSIONS. 39 then db_ d= - and, on writing cos x cos /fx + sinin x in x for cos (x cos 0), the integral becomes U U cos x + U2sin x, 1 where U1 = 2 cos x. (2/, - /2)1- d/, U2 = 2 sin taz. (2/u - /2)n- dcL. J In the integral U1 put /Lx = t: then it assumes the form U" = " cos t. tn- dt x= +- L cos t. t'- dt-. V, x (/n+, where V= fx { 2( - -1 t-n cost dt. By breaking up the interval from 0 to x into the intervals (o 7r\ /(7r 37r)\ 37 57r) (2r-l x), where (r-1) is the greatest multiple of wr contained in x, it may be proved that V is finite, however large x may be; hence the most important term in U1 is 2 + 1 rx -_- I tn- costdt. This represents the true value more and more nearly the larger x becomes, and at the same time approaches the value 2f+f 2n+l (2n + 1)?r 2o tr- cos tdt=, + (n - 2) cos 2 )7 xn+:Jo XnW+ 2 4 (supposing that x is positive). In the same way it may be proved that the approximate value of U2 when x is very large and positive is 2+ I (2n + 1) 7r "+t II (-? ~,)/ sin 4 xn+1 (n2 4~~~~~~~~~~ 40 SEMICONVERGENT EXPANSIONS. [IV. and hence, to the same degree of approximation, Xn Jn (W)n = /~n, (n - ) { U cos + U sin 4 = -- cos - 2. 9 /2 (2n + 1)r ) - COS 9I V TX [ 4 Comparing this with 87 we see that /2 (2n + 1) 7r C= - a 7 rr 4' and hence that xfn (x)A= / P cos (2 + 1) 7r +Q sin(2n + 1) 7r 92 wih P- 1 - (4n2 1) (4n2- 9) with P=l 2-18 — ) 2! (8x)2 (4n2 - 1) (4n - 9) (4n2 - 25) (4n2 - 49) 4! (8X)4 42 - 1 (4n2 - 1) (4z2 _ 9) (42' - 25) 8x +.... - -8 3!(8x)3 For the sake of reference it is convenient to give in this place the corresponding expression for Y,, which may be deduced from a result of Hankel's (Math. Ann. I. (1869), pp. 471, 494) to be considered later on. If we write y for Euler's constant, which is otherwise denoted by - #(0) or -I'(0) II(0), and the value of which is ry=-(0)= '57721 56649 01532 86060 65... 93 and if log 2 is the natural logarithm of 2, the value of which is log2 = 69314 71805 59945 30941 72..., 94 then the approximate value of Y, is, when x is a large real positive quantity, Yr (x) = (log 2 - y) J, (x) + v [Q cos { (f - +1 -Psin — { 4 )-x}, 95 SEMICONVERGENT EXPANSIONS. 41 the values of Jn (x), P, and Q on the right hand being those given above. Thus to the first degree of approximation Y7 = y2 X(log 2 - r) cos (2n + 1) r 7_ 4rsin {(2n + 1) -r_ } - /si 2n+ 7T =(log 2 - ) J, + Jn+. 96 The value of log 2 -7 is, to twenty-two places, log2-7y=-11593 15156 58412 44881 07. 97 It is interesting to confirm these results by means of the relation Jn+' Y - J, Yn+i = - The approximate formulae for J, and Yn hold good for all values of x with a large modulus, provided the real part of x is positive, and that value of /x is taken which passes continuously into a real positive value when x is real and positive. The case when x is a pure imaginary will have to be considered separately in a subsequent chapter. We will now briefly consider the case, already alluded to, when n is the half of an odd integer. By the general definition of Jn (x), we have - cos x, V 7TX, _ x2 4 ) X~ X2 X4 and 2=2nII(~ ) 1-2-. +2.3 4.3.5 -"4 /2 sinx / 2 =A/- iv — = /-sin X. V X x V 7rX Hence and by means of the relation j+- 2rn j - Jn+l = -- in - J?1-1 XC 42 SEMICONVERGENT EXPANSIONS. [IV. we may calculate the expression for Jk+2 where k is any positive or negative integer. The functions thus obtained are of importance in certain physical applications, so that the following short table may be useful. 2n Jo x \/17X 2 r 1 sin x sin x 3 - cos x x 5 1 -1sin x - os x X2 X 7 (15 6). -15 )945 420 15\. 945 105 11 - - +- sin - ---- -+ 1 cos x \5 x3 x I 41cos x2 - i cosx COS X - 3 - sin x - w - 5 -inm + -1 cosX x \X2 I /15 \. (15 6\ -7 - - ) sin - - - ) cos \x2 x3 x/ /105 10>. (105 45 c,\ - 9 (-3 - ) sin x + (x4- 4 + 1 cos x x3 X X4 X2 I /945 105 -. /945 420 1.5\ -11 -V 4 -- + i sinx- - - + cos ~ 4 X2 ) 5 X3 X ) These expressions may be derived from 92 by assigning the proper value to n; the series P and Q terminate when 2n is an odd integer, and we thus see how it is that the analysis employed in finding the approximate value of J, leads to the exact value in this special case. The reader who is acquainted with the modern theory of linear IV.] SEMICONVERGENT 'EXPANSIONS. 43 differential equations will observe that the "indicial equation" corresponding to d2u 1du (1 _ n) dx2 dx xl is r2 - n2 = 0; so that when 2n is an odd integer we have the case when the roots of the indicial equation, although not themselves integral, differ by an integer; and it is this circumstance that gives rise to the exceptional character of the solutions. (Cf. Craig, Theory of Linear Differential Equations, vol. I. (1889), chap. 5.) CHAPTER V. THE ZEROES OF THE BESSEL FUNCTIONS. FOR the purpose of realising the general behaviour of a transcendental function it is important to discover, if possible, the values of the independent variable which cause the function to vanish or become infinite or to assume a maximum or minimum value. It has already been observed that the function J, (x) is finite and continuous for all finite values of x; we will now proceed to investigate the zeroes of the function, that is to say, the values of x which make J~ (x) = 0. It will be supposed in the first place that n is a positive integer, or zero, this being by far the most important case for physical applications. On account not only of its historical interest, but of its directness and simplicity, we reproduce here Bessel's original proof of the theorem that the equation J, (x)= 0 has an infinite number of real roots*. It has been shown (see 46 above) that 1 f7r J (g) = - cos (x sin lb) df 7'JO = 2 f1 o it = - cos (xt) dt 2m + rn' Suppose that x= — 2 - r where m is a positive integer, and m a positive proper fraction; then J () 2 [1 (2m + mr) rt dt 7'Jo 2 V/_t2 2 f2fm+' Trv dv -- -- COS - ~ )2 -v 7rJ 2 V{(2m + m')2 - v' ' * Berlin Abhandlungen (1824), Art. 14. THE ZEROES OF THE BESSEL FUNCTIONS. 45 Jb 7TV dv Now co2 { (2m + m')2 - V2} rb-h (ho ir\ du J Cosh T 2) V{(2m + m')2- (h + U)2}' and hence, by taking a = h -1, b = h + 1, and putting h successively equal to 1, 3, 5,... (2m - 1), we obtain, on writing /J for 2m + m', 2n /, 2 f 7_ -1 1+ Jo ( 2) f- sin - + _ 7r, 2 ( _(i 1U)2 42 -(3 +u)2 (- 1)m m T 2 1)~. fo'~' 2r +- (- cos - 7r o p2 -2 _ (2m + u)2 The integral +'1. 7ru du fJ_ sin 2 d, a =Jsin 2 {V/,2-(h+u)2 -// -Vy }2 d,) and is therefore positive; moreover oB " 7,t du fJo - 2 \/_ (2rm + t)2 is evidently positive. Therefore Jo (x) has been reduced to the form J (x) = - I1 + 2 - u3 +... + (-) 2m, where ul, u2,... u, are a series of positive quantities and 1< U < <3... < Z. Therefore J (x) is positive or negative according as m is even or odd; and consequently as x increases from krr to (k + 1) r where k is any positive integer (or zero) J (x) changes sign, and must therefore vanish for some value of x in the interval. This proves that the equation J (x) = 0 has an infinite number of real positive roots; the negative roots are equal and opposite to the positive roots. It has been shown in the last chapter that the asymptotic value of J0 (x) is /2 / 7r V7rX \ 4/ 46 THE ZEROES OF THE BESSEL FUNCTIONS. [V. so that the large roots of Jo (x) are approximately given by = (k + a) vT, where k is a large positive or negative integer. To give an example of the degree of the approximation, suppose k = 9; then (k + ) -r= 30-6305..., the true value of the corresponding root being 30-6346.... It has been proved that dJ0 (x) J1 (X) = d dx now between every two successive roots of Jo (x) the derived function Jo (x) must vanish at least once, and therefore between every pair of adjacent roots of J0 (x) there must be at least one root of J, (x). Thus the equation J1 (x) = 0 has also an infinite number of real roots; we infer from the asymptotic value of J1 (x) that the large roots are approximately given by =(k + ) 7r, where k, as before, is any large integer. Now let n be any positive integer, and put (a>f J. (x) =R, 2 4 'b? then it may be verified from I6 or io that dRn so that Rn+, vanishes when R, is a maximum or a minimum. We infer, as before, that between any two consecutive roots of Rn = 0 there must be at least one root of R,,+, = 0; and hence, by induction, that the equation J, (x) = 0 has an infinite number of real roots. The existence of the real roots of Jn(x) having thus been proved, it remains for us to devise a method of calculating them. We will begin by explaining a process which, although of little or no practical value, is very interesting theoretically. Let ~ and Rn have the meanings just given to them, and put Rn = fs (S)+ (+ )O THE ZEROES OF THE BESSEL FUNCTIONS. 47 where f, (|) is the sum of the first s terms of R,, and js (~) is the rest of the series. Then when s is very large fs (~) is very small for all finite values of I, and therefore, if f3 is a root of R,, f (B) must also be very small for a sufficiently large value of s. For large values of s, therefore, there will be real roots of the algebraical equation f () = 0 which are approximations to the roots of R=O;0 and even for moderate values of s, we may expect to obtain approximations from f, ()) = 0, but we must be careful to see that ps (/3) is small as well as fs (/) = 0. If the real roots of the equations f = 0 are plotted off, it will be found that they ultimately fall into groups or clusters, each cluster "condensing " in the neighbourhood of a root of R, = 0. The points belonging to a particular cluster are, of course, derived from different equations f = 0. As an illustration, suppose n = 0; then t2 t2 + 4 0 ~ + 12.22 12. 22. 32 12 22. 3242- *' and the equations fs = 0 are -2_4+4=o 3 _ 92 + 36- 36 = 4 - 16 3 + 144:2 _ 576( + 576 = 0, and so on. The real roots of these are, in order, 1 2, 2 1-42999... 1-44678..., 5-42... and we already see the beginning of a condensation in the succession 1-42999, 1-44678. If we put - = 1-44678, we find x = 2405...; and the least positive root of J0 (x) = 0 is, in fact, xI = 2-4048... so that w = x, to three places of decimals. 48 THE ZEROES OF THE BESSEL FUNCTIONS. [V. The best practical method of calculating the roots is that of Stokes*, which depends upon the semiconvergent expression for Jn (x). To fix the ideas, suppose n = 0. Then we have approximately () =/ {P cos( - ) + Q sin( - 12.32 12.32 52 72 where P = 1 - +, 2!(8x)2 4!(8x)4 1 12.32.52 12.32. 52 72. 92 Sw8 3!(8x)3 + 5!(8x)" Put P=Mcos, Q=Msintr; then M=/P2 + Q2, =tan-1 Q Now it is not difficult to see that when x is so large that we may use a few terms of P and Q to find an approximate value for J, (x), we are justified in calculating VP2 + Q2 and tan-, Q as if P and Q were convergent series; the results, of course, will have the same kind of meaning as the series P and Q from which they are derived. We thus obtain the semiconvergent expressions M 1- l1 + 53 M 162 +512x4 y.a, (/1 33 3417 98 = tan- 8-x 512 + 16384x5 ' )J The value of Jo (x) is (approximately) 2/ z (- 4 )' 99 V 2 Mcos X - - 99 which vanishes when = being any integer.+ kc being any integer. Write, for the moment, 0 = ( - 4) 7T; * Caumb. Phil. Trans. ix. (1856), p. 182. V.] THE ZEROES OF THE BESSEL FUNCTIONS. 49 then we have to solve the transcendental equation (/1 33 + 3417 ) = + tan-l8 - 33 + 1 4i 7 qb +tan- 8x 512x3 16384x'-' on the supposition that f and x are both large. a b c Assume x=+ bc+ 3+ +. -; then, with the help of Gregory's series, abc a b c + +'" 1 /l a \ 33 / 1 \ =+851203 3.512 3 * +... 1 31 and therefore a = b= -38 etc. 8' 384' Substituting for b its value (k - 1) r, we have finally x 1 31 7Vg ( 4) 27T2 (4k - 1) 6T4(4k - 1)3 ' or, reducing to decimals, x '050661 -053041 '262051 -= k — 25 + +... I00oo ^r ^41k- 1 (4Ik-1)Y + (4kc- 1)00 The corresponding formula for the roots of J, (x) = 0 is x - '25-151982+ 015399 -245270 = - k+ 25- -+ f 3+.... I00 V 41k+1 (4k+1)3 (4k + 1)5 The same method is applicable to Bessel functions of higher orders. The general formula for the kth root of Jn (x) = 0 is m- 1 4 (m- 1)(7m -31) =a 8a- 3 (8a)3 32 (m- 1) (83m2 - 982m + 3779) IOOo 15 (8a)5 where a = 4= r (2n - 1 + 4), m = 4n2. G.M. 4 50 THE ZEROES OF THE BESSEL FUNCTIONS. [v. This formula is due to Prof. McMahon, and was kindly communicated to the authors by Lord Rayleigh. It has been worked out independently by Mr W. St B. Griffith, so that there is no reasonable doubt of its correctness. It may be remarked that Stokes gives the incorrect value '245835 for the numerator of the last term on the right-hand side of IOI'; the error has somehow arisen in the reduction of 1179/57r6, which is the exact value, to a decimal. The values of the roots may also be obtained by interpolation from a table of the functions, provided the tabular difference is sufficiently small. The reader will find at the end of the book a graph of the functions J (x) and J1 (x) extending over a sufficient interval to show how they behave when x is comparatively large. It seems probable that between every pair of successive real roots of Jn (x) there is exactly one real root of Jn+i (x). It does not appear that this has been strictly proved; there must in any case be an odd number of roots in the interval. With regard to values of n which are not integral, it will be sufficient for the present to state that if n is real the equation J, (x) = 0 has an infinite number of real roots; and if n > - 1, all the roots of the equation are real. CHAPTER VI. FOURIER-BESSEL EXPANSIONS. ONE of the most natural ways in which the Bessel functions present themselves is in connexion with the theory of the potential. This has, in fact, already appeared in the introduction (p. 3); we will now consider this part of the theory in some detail, adopting, in the main, the method of Lord Rayleigh (Phil. Mag. November 1872). If we use cylindrical coordinates r, 0, z, Laplace's equation V2b = 0, which must be satisfied by a potential function E, becomes a20 1 ab 1 a8 a2(6 r2 + r + + r2 s a2 oI0 Assume 0 = ue-K cos nO, 102 where Kc is a real positive quantity, and u is independent of 0 and z. Then b will satisfy Laplace's equation if d2u 1 dqt n, 2 d- + - + K2-o u=0; dr2 r dr r2) I=0; 103 whence t = AJ, (cr) + BJL (Kr)1 or u = AJn (Kr) + BYn(fcr) 04 according as n is not or is an integer. Consider the two particular solutions = eK cos nOJf (Kr) ) 4 = e- z cos n0J, (Xr); 05 4-2 52 FOURIER-BESSEL EXPANSIONS. [VI. then since V2p = 0 and V2I = 0, Green's theorem gives - dS =I- -dS io6 the surface integration being taken over the boundary of a closed space throughout which b and j are finite, continuous, and onevalued; as usual, and de denote the space rates of variation 'alued; as av av of b and '+ along the outward normal to the element dS. First, suppose that n is an integer, and that K and X are positive. Then we may integrate over the surface bounded by the cylinder r = a, and the planes z = 0, z = + co. When z = 0, = cos n0J, (or), ~ = X cos nOJn (Xr), and the part of the integral on the left-hand side of Io6 which is derived from the flat circular end bounded by z = 0, r = a, is 72r ra ra X fIf rdrdr cos2nJ, (Kr) J, (Xr) = Jn(Kr) J (Xr) rdr. When z= + oc, 4 X vanishes, and nothing is contributed to the integral. For the curved surface r = a, e- = e-Kz cos naJ (,ca), so that the corresponding part of the integral is XJn (Ka) J, (Xa)ff adOdze-(KX)z cos2nO o Xa - Jn (Ka) J(Xa). Working out the other side of io6 in a similar way, we obtain x J,, (Kr) J (Xr) rdr + a4 Jn (Ka) Jn (Xa) = a X +Ka = Jn (Kr) Jn. (Xr) rdr + J, (ca) J,, (Xa) K + X FOURIER-BESSEL EXPANSIONS. 53 or, finally ra (c - X) fJn (Kr) Jn (Xr) r dr -- atX {Jn (Ica) Ji (Xa) - K J (Ka) Jn (Xa)}. 107 K+X The very important conclusion follows that if Kc and X are different, ra Jn, (r) J, (Xr)rdr =O, o8 provided that XJn (Kcca) Jn (Xa) - KcJ (fca) Jn (Xa) = 0. IO8' The condition io8' is satisfied, among other ways, (i) if K, X are different roots of J, (ax) = 0, (ii) if they are different roots of J' (ax) = 0, (iii) if they are different roots of AxJ' (ax) + BJ, (ax) = 0, where A and B are independent of x. All these cases occur in physical applications. In the formula I07 put c X + h, divide both sides by h, and make h decrease indefinitely; then we find, with the help of Taylor's theorem, that jJ 2(xr)rdr _= - {Xa [Jn (Ja)]2 - J,, (X a) Jn (a) - XaJ (&a) J'n (Xa )} 2X Reducing this with the help of the differential equation, we obtain finally re *2 ( }?2 J J (\r) rdr = 2 {JP (a) + ( -2a2) J(a}. 109 When n is not an integer, we may still apply Green's theorem, provided that in addition to the cylindrical surface already considered we construct a diaphragm extending from the axis to the circumference in the plane 0 = 0, and consider this as a double boundary, first for 0 = 0, and then for 0 = 27r. 54 FOURIER-BESSEL EXPANSIONS. [VI. When = 0, ~> = e-K" J?,(or),-i = 0; and when 0 = 2r, =e KZ cos 2n7rJn (cr), A - e z sin 2n7rJn (Xr); therefore the additional part contributed to the left-hand side of Io6 is - n sin 2nJr cos 2nr e+X)zn (Kr) J- (Xr) drdz. Now this is symmetrical in Kc and X; therefore the same expression will occur on the right-hand side of Io6, and consequently the formula 107 remains true for all real values of n. In the same way the formulae I8 and Io9 are true for all values of n. To show the application of these results, we will employ the function = e- Z JO (Xr) to obtain the solution of a problem in the conduction of heat. Consider the solid cylinder bounded by the surfaces r = 1, z = 0, z = + o, and suppose that its convex surface is surrounded by a medium of temperature zero. Then when the flow of heat has become steady, the temperature V at any point in the cylinder must satisfy the equation V2V= and moreover, when r = 1, Or where k is the conductivity of the material of the cylinder, and h is what Fourier calls the "external conductivity." If we put V= qb, the first condition is satisfied; and the second will also be satisfied, if xcJo (X) + hJo (X) = 0. 1T Suppose, then, that X is any root of this equation, and suppose, moreover, that the base of the cylinder is permanently heated so VI.] FOURIER-BESSEL EXPANSIONS. 55 that the temperature at a distance r from the centre is Jo(Xr). Then the temperature at any point within the cylinder is V= e-z J, (Xr), because this satisfies all the conditions of the problem. The equation IIo has an infinite number of real roots X\, X2, etc. so that we can construct a more general function =, Ae- Jo (Xr) II 1 and this will represent the temperature of the same cylinder when subject to the same conditions, except that the temperature at any point of the base is now given by <o= AsJo(\Xr). 112 1 Now there does not appear to be any physical objection to supposing an arbitrary distribution of temperature over the base of the cylinder, provided the temperature varies continuously from point to point and is everywhere finite. In particular we may suppose the distribution symmetrical about the centre, and put qo =/(r) where f(r) is any function of r which is one-valued, finite and continuous from r= 0 to r=l. The question is whether this function can be reduced, for the range considered, to the form expressed by 112. Assuming that this is so, we can at once obtain the coefficients As in the form of definite integrals; for if we put f (r) = AsJo (Xr) 113 it follows by 107, o19, and I Io that Jo(sXr)f (r) rdr = As, JO (Xfr) rdr 2 2+ 1) J2 (X, and therefore As= (h2 + 2)J (S) JoJo (Xr) f (r) rdr. I 14 (h2+kfo 56 FOURIER-BESSEL EXPANSIONS. [VT. Whenever the transformation I 3 is legitimate, the function = Ase-^sZ Jo (Xsr) I 5 gives the temperature at any point of the cylinder, when its convex surface, as before, is surrounded by a medium of zero temperature, and the circular base is permanently heated according to the law oo =f(r) = =AJo (Xsr); the coefficients A8 being given by the formula 114. A much more general form of potential function is obtained by putting b = E (A cosn0 + B sin n) ez Jn (Xr) 6 where the summation refers to n and X independently. If we restrict the quantities n to integral values and take for the quantities X the positive roots of Jn(X)= 0, we have a potential which remains unaltered when 0 is changed into 0 + 27r, which vanishes when z = + o, and also when r = 1. The value when z = 0 is 0, = (A cos nO + B sin nO ) J~ (Xr). 117 The function < may be interpreted as the temperature at any point in a solid cylinder when the flow of heat is steady, the convex surface maintained at a constant temperature zero, and the base of the cylinder heated according to the law expressed by 17. We are led to inquire whether an arbitrary function f(r, 0), subject only to the conditions of being finite, one-valued, and continuous over the circle r = 1, can be reduced to the form of the right-hand member of I I7. Whenever this reduction is possible, it is easy to obtain the coefficients. Thus if, with a more complete notation, we have f(r, 0) = 2 (An, cos nO + B,, sin nO) Jn (Xr) I18 we find successively f (r, 0) cos nOdO =,An, Jn (Xr) 0 s and f(r, 0) cos nOJn (Xr) rdOdr = rAn, Jn (Xer) rdr 2o (X0), - zJ2( -An, FOURIER-BESSEL EXPANSIONS. 57 by Io9; so that A, s= rJ2( ) f l /(r, 0) cos n tJn (\,r) rd0dr;) and in the same way 18' Bns =J,(x) f(r, 0) sin nJn (X,r)rd0dr. Other physical problems may be constructed which suggest analytical expansions analogous to 113 and 118; some of these are given in the Examples. We do not propose to discuss the validity of the expansions obtained in this chapter; to do so in a satisfactory way would involve a great many delicate considerations, and require a disproportionate amount of space. And after all, the value of these discussions to the practical physicist, in the present stage of applied mathematics, is not very great; for the difficulties of the analytical investigation are usually connected with the amount of restriction which must be applied to an arbitrary function in order that it may admit of expansion in the required way, and in the physical applications these restrictions are generally satisfied from the nature of the case. Such a work as Fourier's Theorie Analytique de la Chaleur is sufficient to show that the instinct of a competent physicist preserves him from mistakes in analysis, even when he employs functions of the most complicated and peculiar description. For further information the reader is referred to Heine's Kugelfunctionen, 2nd edition, Vol. II. p. 210, and to a paper by the same author, entitled Einige Anwendungen der Residuenrechnung von Cauchy, Crelle, t. 89 (1880), pp. 19-39.* As an example of the conclusions to which Dr Heine is led, consider the expression on the right-hand side of I113, where the coefficients As are determined by I 14; then if the infinite series AJo (Xr) is uniformly convergent, its value isf(r). Similar considerations apply to the expansion I 18, and others of the same kind. There are a few elementary arguments, which, * See also the papers by du Bois-Reymond (Math. Ann., iv. 362), Weber (ibid. vi. 146), and Hankel (ibid. viii. 471). 58 FOURIER-BESSEL EXPANSIONS. [VI. although not amounting to a demonstration, may help to explain the possibility of these expansions in some simple cases. Suppose, for instance, that b is a function which is one-valued, finite, and continuous all over the circle r = 1, and admits of an absolutely convergent expansion co 0 where is a homogeneous rational interaly. Then by puttion of degr cos 0, in Cartesian coordinates x, y. Then by putting x= r cos 6, y = r sin 0, this is transformed into v0 = vo + v rr + v,,2 +... = 2s8, o where vs is an integral homogeneous function of cos 0 and sin 0. By expressing v, in terms of sines and cosines of multiples of 0, and rearranging the terms, b may be reduced to the form ( = po + (p, cos 0 + a-7 sin 0) + (p2 cos 20 + 2 sin 20) +.. = I (ps cos sO + a- sin sO), where ps and as are, generally speaking, infinite power-series in r, each beginning with a term in r8. Now if XI, X,, etc. are the roots of Js(X)= 0, the series for J8 (Xir), J8 (X2r), etc. each begin with a term in rS, and if we put ps = A()J (jr) + AJS (X2r) +.. it may be possible to determine the constants A(, A(), etc. so as to make the coefficients of the same power of r on both sides agree to any extent that may be desired. If the result of carrying out this comparison indefinitely leads to a convergent series A()J. (X\r) + A(sJ. (X2r) + 1 uS \'~1 ' j 1.L2 2 o\o we may legitimately write p,= SAm)Js (\mr), and in like manner we may arrive at a valid formula as = B( J, (Xmr); and then Q = E [cos seOA()J8 (X,r) + sin s0JB3lJs (Xr)], s n, m which is equivalent to 1I8. It will be understood that this does not in any way amount to a proof of the proposition: but it shows how, in a particular case of the kind considered, the expansion may be regarded as a straightforward algebraical transformation verifiable a posteriori. CHAPTER VII. COMPLEX THEORY. MANY properties of the Bessel functions may be proved or illustrated by means of the theory of a complex variable, as explained, for instance, in Forsyth's Theory of Functions. A few of the most obvious of these applications will be given in the present chapter. To avoid unnecessary complication, it will be supposed throughout that n + is real and positive, and, unless the contrary is expressly stated, that the real part of x is also positive. Then, as already proved (p. 38), Xn r Jn (x) =.t 7 -/ (n- ) J cos (x cos c) sin2n bdb. Now cos (Xt) (1 - t2)n-. dt cos (x cos b) sin2n dc = cos (t) (1- dt f+l exti(1 - t2)ndt J-1 on the supposition that the integral is taken along the axis of real quantities from - 1 to + 1. We may therefore write xn exzi (1 - z2)n-i dz = 2n V7rTII (n - ~) Jn (x), 19 and this will remain true for any finite path of integration from - 1 to + 1 which is reconcilable with the simple straight path. 60 COMPLEX THEORY. [VII. We are thus naturally led to consider the function u = xn exzi (1 z2)n — dz 120 where a, /3 are independent of x. It will be found that d2u 1 du (/ dx n + ( _ + - 2n + = ^-'f x1 -) (12 2n + 1) iz} ez (1 - Z)n- dz a = _- n- l dd exzi (1 - 2)n+'} dz; therefore u will be a solution of Bessel's equation if exzi (1 - z2)n+ vanishes when z = a and also when z = /. Under the restrictions imposed upon x and n, the admissible values of a and / are +1, -1 and k+ooi, where k is any finite real quantity. By assigning these special values to a and / and choosing different paths of integration, we obtain a large number of solutions in the form of definite integrals. Each solution must, of course, be expressible in the form AJn+BJ-n or AJn +BY,; the determination of the constants is not always easy, and in fact this is the principal difficulty that has to be overcome. It must be carefully borne in mind that, in calculating the value of u for any particular path, the function (1 - z2)n-2 must be taken to vary continuously. The value of u is not determinate until we fix the value of (1 - Z2)2- at some one point of the path. If U is one value of u for a particular path, the general value for the same (or an equivalent) path is t= e(=n-l) k'i U] where k is any real integer. Consider the function u == cxn eXzi ( - z2)n- dz; 121 we may choose as the path of integration a straight line from 1 to 0, followed by a straight line from 0 to oci. Then if t VII.] COMPLEX THEORY. 61 denotes a real variable, and if we take that value of (1 - 2)n-2 which reduces to + 1 when z = 0, U, = x j~'eti (1 - 2)l- dt + i e-t (1 + t2) -4 dt then Xn fcos(sing)cos)n fd/ -ICJ = - n cos ( sin <>) cosa2ndo /o 7r -i ~ {| \ sin ( sin ) osh2l (o d-.1 is lo a sto so value of } For convenience, let us write Cn = 2n7rI (n- 1); 122 r2 then xn cos (X sin 6) cos2n 6 do = 2CGJn, Jo and if we put 2 cn | sin (x sin 6) cos2n 0 df -Xn e-xin1" ' cosh2 d0 =~ CnTn, 123 Jo T7, is also a solution of Bessel's equation, and one value of u, is U =-1 Cn (Jn +iTn); I24 the general value being vU1 = e(2n-i) k7ri 125 where k is any real integer. In the same way, if u2 = xnf exzi (1 - zA dz, 126 taken along a straight line from - 1 to 0, followed by a straight line from 0 to aoi, one value of u2 is U2 Cn (Jn -iTn) I27 and the general value is e(2n1-i )k7i U2 62 COMPLEX THEORY. [VII. Now consider the integral +o0o = xnj exzi(1 - z2)n-~ dz, 128 taken along a path inclosing the points -1 and + 1, and such that throughout the integration z\ > 1, where Izl means the absolute value, or modulus, of z. Then, by the binomial theorem, we may put (1 - z2)- = e(n-) i {zn-l - (n - ) Z2n-3 +... and therefore one value of u, is given by e- (n-e) 7r Us n= (s (_) - (n) (- )... (n + ) [+i Z 2 l =xn 2 exzi 2n-2S-1 d,. s=0 s! J +00o The integral on the right-hand side may be evaluated by taking the path along the axis of imaginary quantities from + ooi to el, then round the origin, and then from ei to + oio. One value of the result is i2n-2S e-xt t2n-2s- dt + e4n7ri i2n-2s e-xt t22l-28-1 dt, which = (-)8 eri (e'ni - 1) e-t t2-28-1 dt fo =(-)8 2ie3ti sin 2n7r (2n - 2s-1) X2n_2s Therefore one value of u, is given by i = 2*4ni~ sin 2n7r (2n-)( -- s + 1) (2n - - 1) X-1 +2S, Observing that (n - ) (n — |)...(n - s + ) (2n-2s - 1) Uls 1 1 (2n-1) I (n-s-1) 22 n (n - 1) (s) and that II (n - s - 1) I (s - n) = 7r cosec (n - s) = (-) r cosec n7r, VII.] COMPLEX THEORY. 63 we have -I (2n - 1) () Xy-~28 q '3= 47r cos n7r elLn I(2n-" 1)S 3I (n-1) 2 22s1 (s) H (8 - n) -ni (-n- ) eei J_ (x) on reduction. The general value is u3 e(21-") k7i 3; and by putting = - 2 we have the special value 2n+l 7rT 3 ^= (n )J-(x) = 2Cn, cos n7r J-n (x), 129 and the general value 3 = e(2n-l) kni U3. It has been convenient to suppose that Iz\ > 1 throughout the integration; but the value of u, will not be altered if we suppose the integration taken along the path ABCDEFGH represented in the figure below. Ocji A H C B G FI D E If the argument of (1 - Z2)1- is taken to be zero along DE, we must take it to be - (2n - 1) 7r along AB, BC, and + (2n - 1) v along FG, GH. Hence we have one value of u, given by '+I u3 = - e- 2(2,-1) 7 l2 + e1 n-1 i U1 + x 1 exti (1 - t)n- dt J — 1 = Cn [I-2e (Jn - iT?) + 2e (Je + IT,') + Jn] = 2CG cos n 7r (J, cos nrr - T, sin n7r). 130 64 COMPLEX THEORY. [VJI. Since n is real, this must agree with the real value of zi3, that is, with 2C0 cos nur J_,. Hence we infer that J-n = Jn cos nr - Tn sin nrr. 131 The formula I23, by which Tn was defined, is intelligible only so long as the real part of x is positive: we may, however, if we like, regard 131 as a definition of Tn for all values of x. Then 123 still remains valid so long as the integrals are finite. When n is an integer, I3I becomes an identity, but the value of Tn is not really indeterminate. To find its real value, write n - e for n, e being a small positive quantity, and n being supposed an integer; then J-(n-e) - (_)n cos e6rJ2_- = (_)n sin 67rTne. Divide both sides by (-)n and make e vanish; thus Lt ()n J-(n-) J(n-e) e=O The expression on the right-hand side may be written in the form ( -n+e n-1 ()n+s 1 (2s (XE, ( f.(e) l~28 2 o H (s) el(- + 6 + ) 2 2 0 6 2 () where (e1 - _ _ 1 (x2-_ /M (n + s) ) (s + ) 2 n (s) n (n + s-e) 2) Let us use the notation (x) = d logn I I( I (x=; I32 then the limit off(e)/ewhen e is zero is IH7 1 (s{ -() og (- + ()-s)} COMPLEX THEORY. 65 Again (p. 12), H (- n + e + s) n ( - e - S - 1) = cosec (n - e - s) (_)n-s-1 r sin e7rr therefore (on-s-I Lt eI (- n +6+)= (n- 1) e=o 1-1 (n - s -]) Finally, then, when n is a positive integer, 7rTn =- ) (I2%0 ITI (s) 2 + - _ x 2S 2 loobe' - 133 + ) I(s). In ) (- + {2)-s) 2 33 Comparing this with the formula 30 (p. 14) we see that, when n is a positive integer, 7rTn = 2 {f, - (log 2 + ~ (0)) J,,, 134 or, with the notation of p. 40, and on substituting for T, its value as a definite integral, Y = (log 2 -y) J, + T, =2,wl (n - 1) (log 2 - y) J cos (x cos ) sin2e b dp Or 2 + wr j sin (X sin l) cos211 doa - 7r e-Z sinh (b cosh2 ) d. 135 When x is a very large positive quantity, the last integral on the right-hand side may be neglected; and then, by applying Stokes's method* to the other two integrals, we obtain the approximate value of Yn given on p. 40 above. There would be a certain advantage in taking Jr, and Tn as the fundamental solutions of Bessel's equation in all cases when n is positive. This course is practically adopted by Hankel, whose memoir in the first volume of the Mathematische Annalen we have been following in this chapter. Hankel writes Y,, for a * Or, preferably, that of Lipschitz, explained later on, pp. 69-71. G. M. 5 66 COMPLEX THEORY. [vII. function we may denote, for the moment, by Y~,; this is defined for all values of n by the formula yn- reni Jn cos n7r - J-n sin 2ntr 3 77eni 136 COS n7T (see 131 above). Since, however, Neumann's notation is now being generally adopted, we shall continue to use Y, in the sense previously defined. It may be remarked that, when n is a positive integer, Yn = 7T, = 2 {Yn - (log 2 - y),}. 137 Hitherto the real part of x has been supposed positive; we will now suppose that x is a pure imaginary. If in Bessel's equation we put x = ti, it becomes d2u 1 du ( n2 n -d+- a _- _ I+ n 0. I38 dt2 t dt t2K ) U 0. 138 By proceeding as in Chap. II. it is easily found that, when n is not an integer, the equation has two independent solutions In (t), I-, (t), defined by tn+2s In (t) = i- Jn (it)- 7Lz+2) (~ 2'242"~(8)n (1 139 t-n+2s I_- (t) = in J_- (it) = ~ 2- (s) n (- + s) one of which vanishes when t = 0, while the other becomes infinite..When n is an integer we have the solution I as before, and there will be another solution obtained by taking the real part of i- Yn (it). But it is found that both these functions (and in like manner In and I_n) become infinite when t = o, and it is important for certain applications to discover a solution which vanishes when t is infinite. We shall effect this by returning to the equation 123, by which Tn was originally defined. If in the expression on the left-hand side of that equation we put x = ti, it becomes int} t- Un + iTT}, COMPLEX THEORY. Gf where Un = f cos (t sinh cp) cosh2f p do, /o "2 rco = =j sinh (t sin b) cos2m b db + sin (t sinh b)cosh2l b db. The function V, is infinite whether n is positive or negative; but if 2n + 1 is negative Un is finite, and it may be verified that tn U, = tn cos (t sinh b) cosh2~n b df is a solution of Bessel's transformed equation 138. Now although this does not give us what we want, namely a solution when n is positive which shall vanish when t is infinite, it suggests that we should try the function t-n f cos (t sinh () u Jo "cosh2 n which is obtained from tn Un by changing n into - n. This function obviously vanishes when t = oo, and it is easily verified that it satisfies the equation 138, so that it is the solution we require. It will be found convenient to write 2K t = rt(- fO~ cos (t sinh;) d4 1( ) I ( —1-i o cosh2' I140 reducing, when n is an integer, to (-)n = ( 1.3 5...(2n-l) t -'nf cos (t sinh b) dc cosh Then In and K&, are solutions of I38 which are available for all positive values of in, and which are the most convenient solutions to take as the fundamental ones, whether or not n is an integer. As in Chapter II. it may be proved that -n = tIn+ n L +l t T'I nI+,,+ + 2n In - In-1 = 0'4 I-I_ 0) 5-2 68 COMPLEX THEORY. [VII. and that the functions Kn are connected by relations of precisely the same form. It is in order to preserve this analogy that it has been thought desirable to modify the notation proposed in Basset's Hydrodynamics, vol. IT. p. 19. The function here called Kn (t) is 2n times Mr. Basset's K/. By applying Stokes's method to the differential equation 138 we obtain the semiconvergent expansion Aet- (4n 2-1) -(n2- 1) (4 4n2 - 9) utAett I - St- + 2!(St) — (4n2 - 1) (4n2 9) (4n2 25) ) 3! (8t)3 +Be- t- + (4n2 - 1) (4n2 - l) (4.2 - 9) t 2! (8t) (4n2 - 1) (42 - 9) (4n2- 25) } + 3! (8st)3+ which terminates, and gives an exact solution, when n is half an odd integer. This may be used for approximate calculation when t is large. When u = K, it is evident that A = 0. A somewhat troublesome and not very satisfactory process, suggested by a formula in Laurent's Traite d'Analyse, t. III. p. 255, leads to the conclusion that B = V2/r. cos nqr, so that the semiconvergent expression for Kn is K /c{r,{ (4n2_ 1) (4n2-_ )(42_-9) 1 Kn cos nqr. e-t I + 8 + - 142 2\,/ +s -t St 2! (8 t)2 The corresponding expression for In is In/r t et ii (4n2 - 1) (4+l2 1) (4n2 -9) 1 in -e I4 - s -.. 143 ^ 2wt (- St 2!(8t)2 There does not appear to be much reason to doubt the correctness of these results, since the formula for In works out very fairly, even for comparatively small values of t, and the formule are consistent, as they should be, with In+l K, - InK,+l = cos n7r. 144 The formula 30 (p. 14) leads us to conclude that if Rn (t) is the real part of i-n Y (it), Ki (t) must agree, up to a numerical factor, with Rn (t)-(log 2 - 7) I, (t); COMPLEX THEORY. 69 this is confirmed by Stokes's investigation, Ccamb. Phil. Trans., vol. ix. p. [38]. It would be easy to multiply these applications of the complex theory to any extent; we will conclude this chapter by giving an alternative proof of the semiconvergent expansion of Jn obtained in Chap. IV. above. The method is that of Lipschitz (Crelle LVI., p. 189), who works out in detail the case when n= 0, and indicates how the general case may be treated. Let us suppose that n + ~ and the real part of x are both positive. Consider the integral t = jexZi (1 - z2)~-2 dz taken, in the positive direction, along the contour of the rectangle whose vertices are at the points corresponding to the quantities - 1, + 1, 1 + hi, - 1 + hi, where h is real and positive. Then the total value of the integral is zero, and by expressing it as the sum of four parts arising from the sides of the rectangle, we obtain the relation 0 = e eti (1 - t2)'- dt + i e+i {1 - (1 + t)i2}1-' dt -+1 _ e(hi+t)i {1 -(hi + t)2}n- dt J -1 - i ex(-l+it)i 1 - (- 1+ it)}- dt, where t is, throughout, a real variable. The first integral is Cnx-"Jn, and when h= 0, the third integral vanishes: hence CnX-1Jn, () = ie-ix (2it + t2)- e- dt fo - iei (- 2it + t2)- e- dt. I45 Jo The argument of 2it+ t2 must be taken to be - when t vanishes: hence (2it + t2) *- = 2 a- e (2iL) t t( - (l + -, \ il 'COMPLEX THEORY. [VII. t where t~'- is real, and the argument of 1 + - vanishes when t = 0. Applying a similar transformation to (-2it + t2)12-, and putting (2n + 1) 7r 4 -=-=, we have 2-b+2 CGx-ltJf (x) = ei e-t t'-' I +(i dt +e- e-x t'-2 ( - ) dt. Now put xt =; then 2- ^+ J,, (x) = ell e I- ( + 2ix)T d + e-" e-~ - (i - di) +B"J;^^(jl- 2iJ"^ —i By Maclaurin's theorem, we may write (]+ 1~ > ) 1 + (n-2) 2 + (n-2)(Th ( 3) ~ 1 2ix 2i = 2! (2ix)2 ++ \/ 2i 2ix/ s! 2- 1! + 2i ' where 0 is some proper fraction, and I - 2- ) may be expanded in a similar way. Substitute these expressions in the preceding equation, and make use of the formula f e- n-.+s d = II (n - 1 + s); then, if we further observe that cl,, = 2 V/7 (2 - ) we obtain V.2 Jn ( ) = cos + -2 4 sin - (2 4.. (to s terms) + R, I46 vrI.. COMPLEX THEORY. where 2( - 2) (n -\ )... (n-+ / 211 (n - 2) R = (2) sI +) 2s! ) 1"-S-d ke | 2ix) I + ) f ( d + ' - 2ix) - -S 2ix1 Each of the integrals on the right hand is increased in absolute value if the quantities under the integral sign are replaced by their moduli: moreover, when s > n -, the values are still further increased by replacing mod (1 + mod (l-BI) '2i 2ix7 each by unity. A fortiori, if we put x = a, - I) -) ) (( 1 rtn (-) (n- ( - )... (-s + ) H (n + s - HSI (iz - ) (2a)s The expression on the right hand is precisely the absolute value of the coefficient of cos or sin, as the case may be, in the (s + 1)th term of the series (-n2 ), - -4) (n - cos ~ + - 4 sin - 2 (-) 4 cos * + csr 2x 21!(2X)2 so that the semiconvergence of this expansion, and its approximate representation of /.. J,, (x) are fully established. CHAPTER VIII. DEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS. MANY definite integrals involving Bessel functions have been evaluated by different mathematicians, more especially by Weber, Sonine, Hankel, and Gegenbauer. In the present chapter we shall give a selection of these integrals, arranged in as natural an order as the circumstances seem to admit. Others will be found in the examples at the end of the book. By the formula 42 (p. 18) or go (p. 38) we have JO (bx) = - cos (bx cos b) dc. Suppose that b is real, and let a be a real positive quantity: then - roe 1 rot r' e-a Jo (bx) dx = dx e- cos (bx cos %) do. Jo7Jo 0 Jo Under the conditions imposed upon a and b, we may change the order of integration on the right hand, and make use of the formula I e-M cos nx dx = — 2 thus f e-ax J (bx) dx = 7o a 2ro+ b2 COS2 ^ Joa2 + b2' This result will still be true for complex values of a and b provided that the integral is convergent. Now if a a, + ai, b = b1 + b2i, VIII.] DEFINITE INTEGRALS. 73 the expression e-~x Jo (bx), when x is very large, behaves like e-aix cosh b2x ' (x), -P) (- ), ~x where P (x) is a trigonometrical function of x; hence the integral is convergent if a,1 [b2I and under this condition we shall still have oo 1 e-a Jo (bx) dx =ab, I47 that value of 1a2 + b2 being taken which reduces to a when b = 0. When b is real and positive, we may put a = 0, and thus obtain f00 1 Jo (bx) dx = 148 o ob' and as a special case.00 ( Jo(x) dx =. 149 Again let b be real and positive, and put ai instead of a, a being real and positive; thus r00"~ 1 e-axi Jo (bx) dx = - If b2 > a2 the positive value of /b2 - a2 must be taken; if b2 < a2 we must put Vb2-a2 = + i Va - b2. We thus obtain Weber's results f^ 1 1 J Jo (bx) cos a dx = O = Jo (ba) sin ax dx = 0 K Jo (bx) cos ax dx = 0,~1. a2 > b2. 151 Jo (bxs) sin ax dx= - J o ~ -_-_y2 2 74 DEFINITE INTEGRALS. [VIII. When a2=b2 the integrals become divergent; the reason is that they ultimately behave like f00 dco JO VX 0f cos (a-ax) dx instead of like c ( - x The formula I47 with which we started is due to Lipschitz (Crelle LVI.); another due to the same author is H (m-1) (- 2+1) (n -_1) X' -sin Mr7, 15 2 J m-1 Jo (ax) dx = - sin r, 15 2 J ~o ~27ro am2 where rn is a positive proper fraction. To prove this we observe that 7T.00 2 2~~ r X I,-1 Jo (ax) d = dx xm- cos (ax cos 0) do, and on changing the order of integration this becomes 2n (m- 1) cos - 2 2 d, raam Jo cosm 0 II (- 1) Cos- 2 1 +1__ which = 2 (1-)- wai (I t)- dt II (m -- 1) cos — II r (2 I) r2 2 2 Now ( 2 )IIm(2-r1\ cosec -2 2 and I ( 2\ =A (see p. 12); thus the expression reduces to DI7V o m+l r m m7h ) r ( 2 + II (2 — 1) sin-27r 1(m-1) cosv / r 7 2, 2 -- -, i,~,. DEFINITE INTEGRALS. 75 that is, to n(MII - \- + 1-) T (m - - 2 rsin mar, 27ramsn the value given above. The result may be written in various other forms: thus if we make use of the formula n (n -1) n ( —2 cos 22 ( - 1 it becomes /. 00 -2 -1 {lI (A} sin -L x n- ( Jo (ax) dx = 2 sin- - 2"n1 II ( -1) I53 am-1 ( 1) 2~^~W reducing, as might have been expected, to l/a when m = 1. We will next consider a group of integrals which have been obtained by Weber (Crelle LXIX.) by means of a very ingenious analysis. Let V be a function which is one-valued, finite and continuous, as well as its space-flux in any direction, throughout the whole of space, and which also satisfies the equation V2+ m2V-= 0. Using polar coordinates r, 0, 9, and putting cos = p, this equation is i. a /2v\ay a / 2) ay\ I a2."n7) M21 r2 (r d-r) + ( _ ' / t,)/ 1 - /A2 S j r+i r+7 Let o - = J Vd/ do; then, observing that d d -1 (I -2)1 | add a((l- I) l _ 2 =0 J~~ -1I-T ( / A ~ 76 DEFINITE INTEGRALS. [VIII. 3v av because V and a are one-valued, we have _ 2I 1 a( 2a) r2 dr \dr/ or, which is the same thing, 82 r a (r) + m2rw = 0; whence o = - (A sin mr + B cos mr), where A and B are independent of r. If co is finite when r=O we must have B= 0, and O0 sin mr mr where to is the value of o when r = 0. Now from the definition of o) it is clear that, if Vo is the value of V when r = 0, r+1 r+7r to= Vo J df do= 47Vo -1 -7r 47r no sin mr and therefore c =-. 154 m r Now consider V as a function of rectangular coordinates a, b, c and put V= (a, b, c); moreover let us write f f /: (f:) (a, b, c) e-21 (a-X)2+(b)2+(c-Z)2} dadb dc =. 155 Then if we introduce polar coordinates by writing a - x = r sin 0 cos q, b - y =r sin 0 sin p, c -z = r cos 0, cos 0 = L, we have f = r2e2r2 dr f f 'd do, where 4' is the transformed expression for (D. DEFINITE INTEGRALS. 77 By the preceding theorem this is 4 = 47r l' J re-P'2 sin mr dr 47r m -m = (x, y, z). V e 4p2. Hence, finally, - 00 -00 c- o,72. m2 - e 4P () (x, y, z). I56 Suppose now that rp is independent of c; then since e-2(c-)2 dc = f e-p't2 dt = V/r/p we have f j:q:) (a, b) e-p2{(a-x)2+(b-y)2} d db = - e p) (, y), 57 -00p -00 P57 the equation satisfied by Q being -2 - - I + m2~ = 0. ~ ab2 In particular we may put ^ = Jo (mr), where r2 = a2+ b2, a=rcos 0, b=rsin0, and suppose that x =y = O; then the formula becomes, after integrating with respect to 0, re-p2 Jo (mr) dr = e 4p. 158 More generally, by putting < = J,, (mr) (A cos nO + B sin nO), x = p cos /, y = p sin /3, we obtain e-P2p f re-p r Jn (mr) dr e2Pprcos (o-) (A cos n0- B sin nO) dO J~O J -7r 7- m2 = e-4p2 J, (mp) (A cos n/3 + B sin n,8). 159 p2 78 DEFINITE INTEGRALS. [VIII. In this formula put A=l, B=i, 3=r}7r; then the integral with respect to 0 is r+7r e2P2pr sin O+ni dO r-7r =2 fcos (2ip2pr sin 0 - nO) dO = 27rinI (2p2pr). The formula thus becomes, after substitution, and division of both sides by 27rine-P2, fnreP212 1 2 +P2P2 re- p' Jn (mr) In (2p2pr) dr = -2 2 e-4p PJn (nip), or, more symmetrically, putting X for m, and /u for 2p2p, rer I 2-1 X 2 X /I Are- J (Xr) In (/,r) dr = 2 e- p J (,/ ) n16o or again, changing /u into i/u, which does not affect the convergence of the integral, 1* X2+~12 re-P frn (Xr) Jn (/r) dr = 2p e 4,2 In. i6 By making, infinitesimal, we obtain the additional result rn+1 e-p2r Jn (Xr) dr = (2) e-4.2 162 In all these formula the real part of p2 must be positive in order to secure the convergence of the integrals. After this singularly beautiful analysis, Weber proceeds to evaluate the integral.00 lq-n-i Jn (Xr) dr which is convergent so long as 0< q < n +. It is known that 1 1 r - _ -+2-q Jn-7 x -(xq e — ) dx; rn+2 —q n (7- )J o DEFINITE INTEGRALS. 79 therefore rq-n- Jn (Xr) dr = (n- 4 drJ n-q rn+1 Jn (Xr) e-2'x dx. Now it may be shown (1. c. p. 230) that the value of the double integral is not affected by changing the order of integration; if we do this, and make use of I62, we obtain 00 GO X,,n r0 k 2 rq — J (Xr) dr 2n+1 I (n - ) e4x — 1 dx Xn 2q -lQq - 1) 2n+l II (n - q) n ( 2q-n-l Xn-q 1I 1( -1) 6 iT ( n_ - ) I63 By putting q = n + 1, we obtain J n (Xr)dr=; 164 and by putting q = n, which is permissible so long as n is not zero, we find: J (Xr) dr =-1. 165 o r n Weber's results have been independently confirmed and generalised by Gegenbauer and Sonine, especially by the latter, whose elaborate memoir (Math. Ann. xvI.) contains a large number of very elegant and remarkable formulae. Some of these may be verified by special artifices adapted to the particular cases considered; but to do this would convey no idea of the author's general method of procedure, which is based upon the theory of complex integration. The formulae obtained are connected by a very close chain of deduction, so that an intelligible account of them would involve the reproduction of a great part of the memoir, and for this we have not space. We are therefore reluctantly compelled to content ourselves with referring the reader to M. Sonine's original memoir. We will now consider a remarkable formula which is analogous to that which is generally known as Fourier's Theorem. It may be stated as follows: 80 DEFINITE INTEGRALS. [VIII. If ) (r) is a function which is finite and continuous for all values of r between the limits p and q (p and q being real and positive), and if n is any real positive integer, then f dXfXp(p)) Jn(Xp)Jn (Xr) dp= 0, if r>pr< q. 66 A very instructive, although perhaps not altogether rigorous, method of discussing this integral is given in Basset's Hydrodynamics, vol. II. p. 15; we will begin by giving this analysis in a slightly modified form. Taking cylindrical coordinates p, 0, z let us suppose that we have in the plane z=0 a thin material lamina bounded by the circles p= q and p =p; suppose, moreover, that each element attracts according to the law of gravitation, and that the surface density at any point is expressed by the formula of= ) (p) cos nO0. Then the potential of the lamina at any point (r, 0, z) is V=fl f7 [+2r pf (p) cos nO'dp do' ~0 + r2 + p2 2rp cos (O' - 0)} Put 6' - 0 = 7, r2 + p2 - 2rp cos j = R2; then f '27r pf (p) {cos nO cos n. - sin nz sin nq} dp dr -J Jo I o(z2 R2) = cosnf Pf (p) cOpos dp d?) Now when z is positive 1 =" 1- f 0e-z Jo (XR) dX (see p. 72); hence the potential on the positive side of the plane (see p. 72); hence the potential on the positive side of the plane z=O is rp;27r ^G V1 = cos n0 dp dJ e-A p) (p) cos n JTo (XR) dX. I67 0 0 By Neumann's formula (p. 27) we have J0 (XR) = Jo (Xr) Jo (Xp) + 2 Z Js (Xr) J, (Xp) cos sy, 1 DEFINITE INTEGRALS. 81 and hence, if we asswme that V, remains the same if we change the order of integration, r 0 rp r2,r V1 == cos Jn dX dp Je-~z pC (p) cos ni Jo (\R) drv Jo q o = 2r cos nO dx e dpe-~ pc (p) J. (Xr) Jn (Xp). By making a similar assumption we find that the potential on the negative side of the plane z = 0 is V2 = 27r cos?nO dX dpeaz pb (p) Jn (Xr) Jn (Xp). a2 a v, Assuming that the values of a and -a can be found by differentiating under the integral sign, and that the results are valid when z = 0, we have ( _ a7- 4) 7r cos n0 ( dX dp Xp <j(p)J. (Xr)J. (Xp). I68 Now, if r lies between p and q, the value of the expression on the left hand is 47ro = 47rr (r) cos nO; in all other cases it is zero, except possibly when r = q or r =p. Thus, if we admit the validity of the process by which I68 has been deduced from 167, the proposition immediately follows. With regard to the case when r =q or p it may be observed that just as the equation (iv2 _ n = 47rro [ < r < p] a\ az az /=o is really connected with the fact that the attraction of a uniform circular disc of surface density ac is altered (algebraically) by an amount 4ro- as we pass from a point close to the centre on one side to a point close to the centre on the other, so we may infer that when r = q or p (ia2 _ _I) - 2rcr \ az aZ /S=o because the expression on the left is ultimately the difference in the attraction, normal to its plane, of a uniform semicircle as we pass from a point close to the middle point of its bounding G.M. 6 82 DEFINITE INTEGRALS. [VIII. diameter on one side of it to the corresponding point on the other*. Thus we are led to the conclusion that adx p 0 (p) Jn, (Xp) Jn (Xr) dp = 2 (r) 66 if r=q or p. This is what the analogy of the ordinary Fourier series would lead us to expect. The assumptions involved in the foregoing analysis are not very easy to justify, especially the first one: we will therefore give the outline of a more satisfactory demonstration, derived, in substance, from the memoirs of Hankel (Math. Ann. vIII. 471) and du Bois-Reymond (Crelle LXIX. 82). Consider the integral i = d\a Xp Jn (Xr) Jnr (Xp) dp where n is a positive integer, and r, p, h are positive quantities, with p > r > 0. If p is increased by a small amount dp, the corresponding change in u is rh du = dx IXp Jn (Xr) Jn (Xp) dp} = pap f XJ, (r\) J,, (pX) d hp dp ( p2 {rJ (ph) J ' (rh) - pJ' (ph) Jn (rh)} by 107. Now suppose that h is very large, and let (2~ + 1)7r t =; then we may put /2 J (pi) = / —/ h cos (a - ph), J,' (ph) =, 2 sin(a - ph), * This argument applies to any disc at a point on the circumference where the curvature is continuous. VIII.] DEFINITE INTEGRALS. 83 and similarly for J (rh), J1, (rh); therefore ultimately when h is very large, i hpdp 2 -du = -pd2 {2 r cos (a - p) sin (a- rn) P rh V- rp -p sin (a -ph) cos (a - rh)} _p dp {sin(p - r) h sin (2a - (p + - r) ) ~rr~ p -r ~ p + r ~7rr P-2 + p+r Hence the value of u when h is infinite is given by 1T Lt^ f sin (p - r) hp 7rr2U= Lt ^ -vn ---i p2 dp h=X ~ p-r, I. fP1 cos (p+r)h. + (_H21+1 - CO_( + p2 dp}.. r p+r. In virtue of two theorems due to Dirichlet Lt p2 p== Lt - (t + r)o dsA o- cr r I+ - (X r consequently =rr U = r JQ= I It has been shown by Hankel that this last formula holds good even when q = 0. It may be thought paradoxical that the value of the integral f f Xp J, (Xr) J, (Xp) dp, 6-2 84 DEFINITE INTEGRALS. [VIII. which vanishes when p = r, should be independent of the upper limit p. The fact is that this statement is true only on the supposition that p and r are separated by a finite interval, however small that may be. Thus it may be verified that if the positive proper fraction e be so determined that esin xx r - dxc == 2r 'h.r then Lt i d\ h p Jn, (Xr) Jn (Xp) dp =, A-= ooo r so that the value of the whole integral is obtained by confining the variation of p to the infinitesimal range (r, r + err/h). It should be observed that the expression rh rr+e f dXJ Xp J~ (Xr) Jn (Xp) dp when h becomes infinite and e infinitesimal has no particular meaning unless a relation is assigned connecting the ways in which h and e respectively become infinite and infinitesimal. It may be worth noticing that these results are analogous to the reduction of the effective portion of a plane wave of light to that of a part of the first Huygens zone. For convenience, let us write r7 J o IXp J. (Xr) J (Xp) X = I (p, ); then it follows from what has just been proved that I if q<r<p, Lt CI (p, h) dp =,, r==q or r = p, [0,, r<q or r>p. In the first case of this proposition the inequalities q < r <p must be understood to mean that r - q and p - r are finite positive quantities; and in like manner with regard to the other inequalities which occur in this connexion. Now suppose that f(p) is any function of p which throughout the interval (q, p) remains finite and continuous and is always increasing or always decreasing as p goes from q to p. Then by a lemma due to du Bois-Reymond DEFINITE INTEGRALS. 85 f (p) 4 (p, h) dp -=f(q) f(J (p, h) dp + {f(p) - f(q) P (p, h) do, where, is some quantity between q and p. Suppose that r is outside the interval (q, p); then, when h becomes infinite, both the integrals on the right hand vanish, and therefore Lt f (p) i (p, h) dp =O [r >p or r < q]. 69 Next suppose r = q; then by the same lemma f (p) (p, h)dp =f(r) f i (p, h) dp + f (p) -f(r)} f (p,h)dp and therefore Lt f(p) ( (p, h) dp = f (r) + M {/(p)-f(r)},, =- 00 r where M is certainly finite, because it is zero if a is ultimately separated from r by a finite interval, and it is I when / = r. Now by I69, if t is a positive quantity greater than p, Lt f/(p) ( () p, h) p=, and hence, by adding the last two formulae, Lt (f (p q p, h) dp = f (r) + M {f(p) -f(r)}. = oo r Since the expression on the left hand is independent of p, the same must be true of that on the right, and consequently M must be zero. Therefore Lt fi(p), (p, h) dp = (r) 170 provided r < p and that r and p are separated by a finite interval. It may be proved in the same way that Lt f (p) (p, h)) dp = f() 170' 7 =o J q when r exceeds q by a finite quantity. 86 DEFINITE INTEGRALS. [vni. Hence, finally, P h f(r) if q<r<p, Lt dp XpfJ(p)J(r)J(p)d), ) =q or =p, O, r<qorr>p, 171 and, as in the other case, we may, if we like, change the order of integration. It has been supposed that f(p) continually increases or continually decreases throughout the interval (q,p); but it may be shown, as was done by Dirichlet in the analogous case of the Fourier integrals, that the formula is still valid if this restriction is removed. The function f(p) may even present any finite number of isolated discontinuities in the interval, and it may become infinite for isolated values of p provided that f(p) dp is finite. It is upon the formula 171 that Hankel bases his proof of the validity of the Fourier-Bessel Expansions briefly discussed in Chap. VI. CHAPTER IX. THE RELATION OF THE BESSEL FUNCTIONS TO SPHERICAL HARMONICS. THERE is a remarkable connexion between the Bessel functions and spherical harmonics which appears to have been first discovered by Mehler (Crelle LXVIII. (1868), p. 134, and Math. Ann. v. (1872), pp. 135, 141). His results have since been developed by Heine, Hobson and others. If, as usual, P,, (x) denotes the zonal harmonic of the nth order, it is known that P (cos 0) = cos" 0 {1 - 2 ( 1) tan2 0 + 0(it- 1)(n - 2) (n - 3)tn - Put 0 = x/n, and suppose that in becomes infinite while x remains finite; then since x Lt cosn -= 1 n=oo n and Lt t tan - = x, n= 00 n it follows that Lt P, T \ - 2 04 Lt P cos = 1- + 2 4 2?=00 \ / = 00 n 2 = J (). I72 This result may also be obtained from the formula 7rP, (cos 0) = I {cos + i sin 0 cos dn d; Jo 88 SPHERICAL HARMONICS. [Ix. for the limit, when n is infinite, of /.. L cos - + i sin - cos b n n, is the same as that of (1 +ix cos ()b, n and this is eixcos; so that 7r Lt P,a (cos ) = fi eicosO d -7rJo (), and hence Lt Pq, (cos - )= Jo (x) as before. Another known theorem (Heine, Kugelfunctionen I., p. 165) is that Q,, (cosh 1 + ___________ n (cosh 0) = 2 J- (cosh 0 + sinh 0 cosh ()l+1 and if we put 0 = x/n, and proceed as before, we find Lt Qa cosh ) = | e- cosh db \?^/ 2 J —oo =-Ko () 173 (see p. 67). Changing x into ix, we obtain Lt P, (cosh - = Io (), \ I74 Lt Qu (cos ) = (log 2 - y) JO (x) - Y( x) (Kugelf I., pp. 184, 245*). Thus every theorem in zonal harmonics may be expected to yield a corresponding theorem in Bessel functions of zero order: it may, of course, happen in particular cases that the resulting theorem is a mere identity, or presents itself in an indeterminate form which has to be evaluated. * There is a misprint on p. 245: in line 8 read C instead of - C. SPHERICAL HARMONICS. 89 Heine has shown that the Bessel functions of higher orders may be regarded as limiting cases of the associated functions which he denotes by P" (x). For the sake of a consistent notation we shall write Pv (x) instead of Heine's P" (x). Then (Kugelf. I. p. 207) 7r Ps (cos 0) - 2HE (n + s) I (T - S) = U2' II (+2) I (cos 0 + i sin 0 cos h)m cos so d, and therefore Lr LtPe (cos ) =Lt {2n (n + s) (nos s)} d.c Now when t is very large we have asymptotically (t)= /27 e-t tt+,2 and therefore to the same degree of approximation, when n is large, 2 n (n - s) II (n + s) 2". 27re-27- (Z + S)l+S+ ( -S)n-s+ II (2nl) /2'r 2n2)+ e-2"l i2th+2 '=/7 (1 + S) ( S hence we infer that t 22L II (n - s) II( + ) (1+s+ (1 - Lt,. - (2n) _5 = L - I1 - 1 =00 17/r nH(2n) nJ oo n n es. e-S = 1. Consequently Lt,/_ P- (cos -) = eixcos cos s8 dc ^oo^TT 7 7)rJo = iJs (x), 175 which is Heine's formula (1. c. I., p. 232). In the same way it may be inferred from the formula (1. c., p. 223) HI (2n + 1) rf cosh sb dc Qu (cosh ) 2 2I (co + s) IH ( - s) o (cosh 0 + sinh 0 cosh Sb)'+1 ' 90 SPHERICAL HARMONICS. [IX. that Lt { 1 Qt (cosh = e-xcosh cosh sf d It= co 71i 7 = (-),K. (x). 176 By changing x into ix we may obtain similar formulae for Is (x) and (log 2 - y) J, (x) - Y, (x). In Chap. VII. it has been supposed that the argument of the functions I (x), K, (x) is a real quantity; in fact the functions were expressly introduced to meet the difficulties arising from the values of J, (x), Y, (x) when x is a pure imaginary. In the.Tugelfmtctionen Heine practically defines his KI (x) by the formula K. (x + Oi) = (- zi) eixc~s8 4 cosh sof d(f) =(-)Ks (-x- Oi), 177 where x + Oi= a (sin a + i cos a), - < a < ir; hence if, to avoid confusion, we write Bs (x) for Heine's Ks(x) and use the latter symbol for an extended definition of the function previously denoted by it, we may write cc KS (x) = (-_)s e-x0o cosh s dc = i-ms U (ix) I78 with x =a (cos a + i sin a), -2 7r< a< 2 7r By putting Ks (- x) = (-)sKs (x) 179 we give a meaning to Ks (x) in all cases except when x is a pure imaginary; and if we define K, (ti), where t is real, by the formula Ks (ti) = {Ks (ti + 0) + Ks (ti - 0)}, we find that Ks (ti) = is {(log 2 - ) J, (t) - Y,(t). I80 From our present point of view, the most proper course is to take as the standard solutions of Bessel's equation, not J,, and Y,,, but J,, and i-n2 K,, (ix) as above defined. The function i- K,, (ix) has in fact already appeared in Chap. VII.; if, when it is made SPHERICAL HARMONICS. 91 one-valued as explained above, we call it G,, (x), then Y,, (x) also becomes a one-valued function defined by Yr, (x) = (log 2 - 7) J,, (x) - G,o (x) s81 with a discontinuity along the axis of real quantities expressed by the formula Y, (t + Oi) - Y, (t - Oi) = - 7ri J, (t). For any point on the axis of real quantities Y, (t)= (log 2 - ) J (t) - G, (). 182 The reader will observe that, in fact, G, () is identical with Heine's K,, (x). We have employed KQ, (x) in a different sense in conformity with the usage now generally current in England: but it must be admitted that Heine's notation is, on the whole, the preferable one. To Heine is also due the following excellent illustration of the methods of this chapter. It is known that, with a proper determination of the signs of the radicals (see Kugelf I. p. 49), P,, (xy - V/ - I. /y2- 1 cos I ) = s (-)SasP8(x) P, (y) cos so, 183 0 where the summation refers to s, and {1. 3. 5...(2s,- )}2 ao = (.. ). s,~ = l s. [' a. 5...(2s,,- 1)},2 (a - s)! (I? s- s) > 01. Now suppose that b c = cos-, y=cos -, n n where b, c are finite real quantities, and that n is very large. Then - 1 = i sin, /y - 1 = i sin n u n and b c b c xy - V/x - 1. / y2 - 1 cos = cos - - +in sin - sin - cos it?t n? n b2 - 2be cos 6 + c2 -1- 1....-... a = cos - n 92 SPHERICAL HARMONICS. [iX. to the second order, if a = Jb2 - 2bc cos + c-2. Making n increase indefinitely, and employing 183, 175, we have 00 Jo J(b2 - 2bc cos + c2) -= A8J, (b) JS (c) cos s, 0 where A Lt{1 3.5..2 - 1)}2 17r (n!)2 22fl 1A - Lt {1.3.5...(2n- 1)}2 nr 2 o=, (n+4s)!(n-s)! 212n Proceeding as on p. 89, we find Lt (2n!)2 n7r L {^/27e-2n(2t)2f+l2w =Z17r Ao= Lt (s )4= Lt -(+ l1; ~=00 24W (n!)4 2~21 {1I/2v e-? ^ 2l+; and similarly -1A, = Lt (2l)2nv- 2As = Lt 24,1 (n!)2 (n + S)! (n -!; thus finally we arrive at Neumann's formula Jo (\/b2- 2bc cos + c2) = Jo (b) (c) + 2 J () J(c) 2 (b) c)cos sb. 184 1 In the same way, from the expansion (n +4 2) Qn {xy -J-l)(y2- ) cos } = ' P6(y) Q() coss 185 o (Kugelf. I. p. 333) we derive the formula Go (/b2 - 2bc cos + c2) = Go (b) Jo (c) + 2 ( Gs (b) J (c) cos so, I86 in which it is supposed that b, c are real and that b > c. By combining 186, 184, and 181 we infer that Yo (/b - 2bccos 0 + c2) = Yo (b) J (c) + 22 YF (b) J, (c) cos so, 187 1 a result originally given by Neumann (Bessel'sche Functionen, p. 65). We shall not proceed any further with the analytical part of the theory; for the extension of spherical harmonics and Bessel IX.] SPHERICAL HARMONICS. 93 functions to a p-dimensional geometry the reader may consult Heine (Kiugelf. I. pp. 449-479) and. Hobson (Proc. L. M. S. xxIn. p. 431, and xxv. p. 49), while for the solution of ordinary differential equations by means of Bessel functions he may be referred to Lommel's treatise, and the papers by the same author in the lMathematische A nnalen. (On Riccati's equation, in particular, see Glaisher in Phil. Trans. 1881, and Greenhill, Quart. Jour. of Math. vol. xvi.) CHAPTER X. VIBRATIONS OF MEMBRANES. ONE of the simplest applications of the Bessel functions occurs in the theory of the transverse vibrations of a plane circular membrane. By the term membrane we shall understand a thin, perfectly flexible, material lamina, of uniform density throughout; and we shall suppose that it is maintained in a state of uniform tension by means of suitable constraints applied at one or more closed boundaries, all situated in the same plane. When the membrane is slightly displaced from its position of stable equilibrium, and then left to itself, it will execute small oscillations, the nature of which we shall proceed to consider, under certain assumptions made for the purpose of simplifying the analysis. We shall attend only to the transverse vibrations, and assume that the tension remains unaltered during the motion; moreover if z =0 represents the plane which contains the membrane in its undisturbed position, and if z= (x, y) defines the form of the membrane at any instant, it will be supposed that 8/8ax and /la:y are so small that their squares may be neglected. Let o be the mass of the membrane per unit of area, and let Tds be the tension across a straight line of length ds drawn anywhere upon the membrane; moreover let dS be an element of area, which for simplicity we may suppose bounded by lines of curvature. Then if r1, r2 are the principal radii of curvature, the applied force on the element is T (+) dS ri r2 VIBRATIONS OF MEMBRANES. 95 and its line of action is along the normal to the element. For clearness, suppose that the element is concave to the positive direction of the axis of z: then the equation of motion is a2z '1 1 a-dS =T (- + - dS cos, at2 \r r2I where Q is the small angle which the inward-drawn normal makes with the axis of z. Now, neglecting squares of small quantities, 1 1 a2z a2z ri r a2 X ay2 and cos r = 1; hence the equation of motion becomes a z / a2z a2z ax2 C, ~y;) ayI T with c2=. 2 It remains to find a' solution of I sufficiently general to satisfy the initial and boundary conditions; these are that z and az/at may have prescribed values when t=0, and that z = 0, for all values of t, at points on the fixed boundaries of the membrane. By changing from rectangular to cylindrical coordinates the equation I may be transformed into a2Z at2Z laZ Ia2Za\ at2 = r +2r ar +r2 O2. 3 Now suppose that the membrane is circular, and bounded by the circle r=a; then we have to find a solution of 3 so as to satisfy the initial conditions, and such that z = 0, when r = a, for all values of t. Assume z = t cos pt, 4 where iu is independent of t; then putting Ph-=Ks, 5 'u has to satisfy the equation a2lt 1 at 1 -a2. 2 -1 — - - q+ /lo t =0; 6 a)-2 - or r2 ac0 96 VIBRATIONS OF MEMBRANES. [x. and if we further assume that u = v cos nO, 7 where v is a function of r only, this will be a solution provided that dv + dv (2- V =. 8 dr2 r dr r v=. It will be sufficient for our present purpose to suppose that n is a positive integer; this being so, the solution of 8 is v = AJn (Kr) + BYn (Kr). From the conditions of the problem v must be finite when r = 0: hence B = 0, and we have a solution of 3 in the form = A Jn (Kr) cos nO cos pt = AJn (Kr) cos nO cos Kcct. 9 In order that the boundary condition may be satisfied, we must have J. (Ka) = 0, IO and this is a transcendental equation to find K. It has been proved in Chap. V. that this equation has an infinite number of real roots /K, K2, K3, etc.; to each of these corresponds a normal vibration of the type 9. The initial conditions which result in this particular type of vibration and no others are that when t =0, z = AJn (Kr) cos nO, -= 0 at By assigning to n the values 0, 1, 2, etc. and taking with each value of n the associated quantities 1^), K(I), C) etc. derived from Jn (ca) = 0, we are enabled to construct the more general solution z = E (An cos nO cos K^) ct + Bns sin no cos Kl ct + Cn cos nO sin Kt?) ct + Dns sin nO sin K() ct) J, (K(?) r), I where Ans, Bns, CGs, Dns denote arbitrary constants. If the initial configuration is defined by =f(r, 0) we must have f(r, 0) = - (A.n cos nO + BA. sin nO) J,, (K'n) r), 12 VIBRATIONS OF MEMBRANES. 97 and whenever f(r, 0) admits of an expansion of this form the coefficients An, B,, are determined as in Chap. V. (pp. 56, 57) in the form of definite integrals. In fact, writing Kg, for convenience, instead of Kc2), is - _ O- f o(r, 0) cos n0fJ, (Kr) rdr 2 fer fcoI3 B-t= 2 Jo f (r 0) Jin0 OJ (Kr, ) sin n ( rdr. Since J,( (cca)= 0, it follows from the formula I9 (p. 13) that Ja, (Ksa) = Jn-l (Kca), so that we may put J,2_1 (Ga) for J' 2(IsCa) in the expressions for Ans, Bi. If the membrane starts from rest, the coefficients Cns, Dns are all zero. If, however, we suppose, for the sake of greater generality, that the initial motion is defined by the equation /az\ () = b (r, 0), we must have C (r, 0) = Ecg2) c (C0S cos nO + D,,8 sin nO) Jn (Kcb) r), 14 from which the coefficients COs, Dn, may be determined. From the nature of the case the functions f(r, 0), q (r, 0) are one-valued, finite, and continuous, and are periodic in 0, the period being 27r or an aliquot part of 2rr; thus f(r, 0),-and in like manner qb (r, 0),-may be expanded in the form f(r, 0) = ao + a, cos 0+a2 cos 20 +... + b sin sin + 2 sin 20 +..., the quantities a., b8 being functions of r. The possibility of expanding these functions in series of the form EAsJn (Kr) has been already considered in Chaps. VI. and VIII. In order to realise more clearly the character of the solution thus obtained, let us return to the normal oscillation corresponding to z = Jn (csr) cos nO cos KscCt, 15 IcK being the sth root of Jn (Ka) = 0. G.M. 7 98 VIBRATIONS OF MEMBRANES. [X. Each element of the membrane executes a simple harmonic oscillation of period 27r 27r a /CsC Ks \/ and of amplitude J, (,cr) cos qn0. The amplitude vanishes, and the element accordingly remains at rest, if J,, (C.r) = 0, or if cos n = 0. The first equation is satisfied, not only when r=a, that is at the boundary, but also when K1 K2 KS-1 r-= -a, r - a,... r= -- a; Ks /Cs Ks consequently there exists a series of (s - 1) nodal circles concentric with the fixed boundary. The second equation, cos n0 = 0, is satisfied when 0 r 37r _ (4n- )7r 0=2-, 0=2.. 0= 2n1' 2n ' * 2n therefore there is a system of n nodal diameters dividing the membrane into 2n equal segments every one of which vibrates in precisely the same way. It should be observed, however, that at any particular instant two adjacent segments are in opposite phases. The normal vibration considered is a possible form of oscillation not only for the complete circle but also for a membrane bounded by portions of the nodal circles and nodal diameters. It is instructive to notice the dimensions of the quantities which occur in the equations. The dimensional formula for 0- is [<] = [ML-2]; that for T is [T] = [MT-2]; hence by 2 that of c is [c] = [LT-1]. Since Ks is found from Jn (Ksa)= O, csa is an abstract number, and [KCs] [L-]. VIBRATIONS OF MEMBRANES. 99 Thus the period 27r//8c comes out, as of course it should, of dimension [T]. If we write au for KSa, so that Cs is the sth root of J, (x) = 0, the period may be written in the form 27ra / 2 /rM Is/ T ~s,\/ T' where M is the mass of the whole membrane. This shows very clearly how the period is increased by increasing the mass of the membrane, or diminishing the tension to which it is subjected. As a particular case, suppose n =0, and let,/u = 24048, the smallest root of J(x) = 0; then we have the gravest mode of vibration which is symmetrical about the centre, and its frequency is 2 vr /= / x 678389. Thus, for instance, if a circular membrane 10 cm. in diameter and weighing '006 grm. per square cm. vibrates in its gravest mode with a frequency 220, corresponding to the standard A adopted by Lord Rayleigh, the tension T is determined by / (2 006 x 6784 = 220, whence /220 2 T= (6784) x 157r= 49560 in dynes per centimetre, approximately. In gravitational units of force this is about 50 grams per centimetre, or, roughly, 3'4 lb. per foot. In the case of an annular membrane bounded by the circles r = a and r = b, the normal type of vibration will generally involve both Bessel and Neumann functions. Thus if we put z= A {Jn (ra) - n ( ) cos nO cos KCt, 17 J (fca) Yn (KC) this will correspond to a possible mode of vibration provided that K is determined so as to satisfy Jn, (ta) Yn (lb) - Jn, (Kb) Yn (ca) = 0. I8 7-2 100 VIBRATIONS OF MEMBRANES. [X. It may be inferred from the asymptotic values of J,, and Ye} that this equation has an infinite number of real roots; and it seems probable that the solution z = S2 {A cos nO + B sin n6} {J: r) Yn (Kr2 COS Ct 19 1J, ((a) Y, O (Kt) is sufficiently general to meet the case when the membrane starts from rest in the configuration defined by z=f(r, 0). Assuming that this is so, the coefficients A, B can be expressed in the form of definite integrals by a method precisely similar to that explained in Chap. VI. Thus if we write Jf (Kr) Yn (Kr) J&(I<:a) T (Ka)' it will be found that fdO f (r, 0) ur cos nOdr = LA, 20 f dc f(r, 0) ur sin ndr = LB, J Q J a a where val L a r a r a s l This value of L may perhaps be reducible to a simpler form in virtue of the condition Jn (Ka) Yn (Eb) - Jn (Kb) YL (Kc) = O. For a more detailed treatment of the subject of this chapter the reader is referred to Riemann's Partielle Differentialgleichungen and Lord Rayleigh's Theory of Sound. CHAPTER XI. HYDRODYNAMICS. IN Chapter VI. it has been shown that the expression b = 5 (A cos n + B sin nO) e-^Z J(Xr) satisfies Laplace's equation V20 = 0, and some physical applications of this result have been already considered. In the theory of fluid motion may be interpreted as a velocity-potential defining a form of steady irrotational motion of an incompressible fluid, and is a proper form to assume when we have to deal with cylindrical boundaries. We shall not stay to discuss any of the special problems thus suggested, but proceed to consider some in which the method of procedure is less obvious. Let there be a mass of incompressible fluid of unit density moving in such a way that the path of each element lies in a plane containing the axis of z, and that the molecular rotation is equal to co, the axis of rotation for any element being perpendicular to the plane which contains its path. Then, taking cylindrical coordinates r, 0, z as usual, and denoting by uz, v the component velocities along r and parallel to the axis of z respectively, a (ur) + (vr) = 0, I ar a2 av a= and -- = 2,. 2 ar az 102 HYDRODYNAMICS. [XI. Equation I shows that we may put ur=-, vr* = ' 3 aZ) ar 3 where r is Stokes's current function; thus equation 2 becomes a I a\ + a I a - 2 0. 3r -^- + ~- a -2to=0. 4 ar r ar az r az. When the motion is steady, ~ is a function of r and z; and if we put q2 = ze 2+ v2 so that q is the resultant velocity, the dynamical equations may be written in the form ap a ( 2)2 a - -A(- 2q)-2 =o, ar ar - r ar o, ap+ a ( _ 2 a= whence it follows that o/r must be expressible as a function of -. The simplest hypothesis is -o = rr, 6 where [ is a constant; on this assumption, 4 becomes a (i + a (l ) -2'r =0. 7 ar r ar az r az Now the ordinary differential equation c' (d_2l)r=0 dI dX - 2z r = 0 dr r dr is satisfied by X = r4 + Ar2 + B, where A and B are arbitrary constants; and if we assume /fr = X + pr cos nz, where p is a function of r only, we find from 7 that d2p 1dp (2 1 dr2rr- - n + 2P0 d rf 4- r dr r - the solution of which is p = 0iJ (nar) + D,,K (nr). Finally, then, -I = 1r4 + Ar2 + B + r, {CI1 (uzr) + D,,K1 (nr)} cos nz, 8 fl XI.] HYDRODYNAMICS. 103 where the values of n and of the other constants have to be determined so as to meet the requirements of the boundary conditions. Suppose, for instance, that the fluid fills the finite space inclosed by the cylinders r = a, r = b and the planes z = + h. Then the boundary conditions are a =O az when r = a or b, for all values of z; and Or when z = + h, for all values of r such that a r - b. One way of satisfying these conditions is to make p constant and equal to zero at every point on the boundary. Now if we put =14/9r 2) 9\2 _ 97\2)_ry C 1 ' (nr) _ K, (m) cos nz =4 (r - a2) (r - b2) - nS _ __ _ ___ ) --? 11 (I() K (na)) cosnh this is of the right form, and vanishes for r=a. It vanishes when r = b, provided the values of n are chosen so as to satisfy I, (na) K1 (nb) - 1 (nb) K1 (na) = 0; IO and, finally, it vanishes when z= ~ h if the coefficients C, are determined so that ZC 1 (n) _ (t = ' (r2 - a2) (r2- b2)/r ( 1(na) K, (na)) = r I I for all values of r such that a > r b. Assuming the possibility of this expansion, the coefficients are found in the usual way by integration. The stream-lines are defined by J = const., 0 = const., so that the outermost particles of fluid remain, throughout the motion, in contact with the containing vessel. (The above solution was given in the Mathematical Tripos, Jan. 1884.) 104 HYDRODYNAMICS. [XI. Some very interesting results have been obtained by Lord Kelvin (Phil. Mag. (5) x. (1880), p. 155) in connexion with the oscillations of a cylindrical vortex about a state of steady motion. Adopting the fluxional notation to denote complete differentiation with respect to the time, the dynamical equations of motion, and the equation of continuity, are in cylindrical coordinates ap ar a D a a _ ap- _= - r02 +0+ -- +a + - ar at ar ao aDz ap + a (rO) a(rO) a (rO) pz Dt Dr D' az ' D r a (r0) Da -+-+- -+- = 0. I3 or r rDa Dz It is to be understood that r, 0, z are treated as functions of t, while r, 0, z are supposed expressed as explicit functions of r, 0, z, t; and the density of the liquid is taken to be unity. We obtain a possible state of steady motion by supposing that i=0, z=0, 0= o (a constant), this makes the resultant velocity U-= r, I4 while the pressure is LI = fco2rdr = 1I0 + 2o 2r, 15 o0 being a constant depending on the boundary conditions. Now assume as a solution of 12 and 13 r = p cos mz sin (nt - s), r0=U + u cos mz cos (t - s6),1 z = w sin mz sin (nt - sO), p = II + w cos nz cos (nt - sO), where s is a real integer, m, n are constants, and p, u, w, r are functions of r which are small in comparison with U. Then substituting in 12 and 13 and neglecting squares and products of small quantities, we obtain the approximate equations dr (n- sw) p - 2u, 8W 17 - - = - (n - so) i + 2wp, mp = (p - sI) W, dp p su dr + - +-+mw=0. i8 dr r r HYDRODYNAMICS. 105 From equations 17 we obtain so (nso)dw 2so }j (n - so) {(n - sw) - 2 w} m {4t)2 (n -st))21, >19 (n - sWo) 2o dr - d -- w9 { [dr r } m {4f2 - (n - S))2} and on substituting these expressions in I8 we find, after a little reduction, d2w 1 dw m2 42 2 (n - So)2} S2) dr2 r dr (n - so)2 r2 If the quantity mn2 {4f2 - (n - sW)2} (n - so)2 is positive, let it be called K2; if it is negative, let it be denoted by - X2. Then in the first case w= A Js (Kr) + BY8 (Kr), 21 and in the second case w = 01C (Xr) + DKs (Xr). 22 The constants must be determined by appropriate initial or boundary conditions. For instance, suppose the fluid to occupy, during the steady motion, the whole interior of the cylinder r = a. Then in order that, in the disturbed motion, w may be everywhere small it is necessary that B = 0 in 20 and D = 0 in 21. To fix the ideas, suppose n, n, s, Wo assigned, and that 4f2 > (n - So)2; then by 19 and 21 A (n - so) {(n - so) KJ: (cr) -2sW J. (/cr) P enm {42 _ (n _ S)21 23 By I6 the corresponding radial velocity is = p cos mnz sin (nt - sO) and if po is the value of p when r =a, the initial velocity along the radius, for r = a, is -po cos mz sin sO. 106 HYDRODYNAMICS. [XI. Now po may have any (small) constant value; supposing that this is prescribed, the constant A is determined, its value being, by 23, A _, {4o2 - (n - sW))21 mp ~~~= K2PTo KJ(Ka) - 2sw 24 J ()n) - J (-ca) (n - s)aJKa) Of course, the other initial component velocities and the initial pressure must be adjusted so as to be consistent with the equations 16- I9. There is no difficulty in realising the general nature of the disturbance represented by the equations I6; it evidently travels round the axis of the cylinder with constant angular velocity n/s. When o is given, we can obtain a very general solution by compounding the different disturbances of the type considered which arise when we take different values of m, n, s; according to Lord Kelvin it is possible to construct in this way the solution for "any arbitrary distribution of the generative disturbance over the cylindric surface, and for any arbitrary periodic function of the tine." The general solution involves both the J and the I functions. Another case of steady motion is that of a hollow irrotational vortex in a fixed cylindrical tube. This is obtained by putting i=0, i=0, r20=c, where c is a constant; the velocity-potential is cO, and the velocity at any point is U=-. 25 r If a is the radius of the free surface, the pressure for the undisturbed motion is I=Io = 2 (+ 2 -) 26 XI.] HYDRODYNAMICS. 107 Putting these values of U and IT in equations I6 and proceeding as before, we find for the approximate equations corresponding to 17 and I8 dr ( cs\ 2c_ \ dr \ r2 * r2' _d-= n-Cs P —, n —r U2 2 c dr r r - - += - -w = 0. dr r r 27 28 Hence sw 1 / cs U = -,n - W mr m 7 2 -- - r 29 1 dw P= mdr' ) and therefore the differential equation satisfied by w is d2w Idw 8f2\ +- - m + 2- I == 0. dr r dr - r2 Consequently w = AIs (mr) + BK] (mr), where A, B are arbitrary constants. 30 If the fixed boundary is defined by r = b, we must have r = 0 when r =b; that is, by I6 and 29 dw = 0 when r =b. dr Thus JIs (mr) Ks (mr)\ IA (mb) K' (mnb) 3I We have still to express the condition that p = IT at every point on the free surface for the disturbed motion. To do this we must find a first approximation to the form of the free surface. In the steady motion, the coordinates r, z of a particle of fluid remain invariable and C ct 0= whence 0= -. In the disturbed motion r does not differ much from its mean value r0, and if we take the equation r= p cos nmz sin (t - sO) 108 HYDRODYNAMICS. [XI. we obtain a first approximation by putting 0 = t, giving p its mean value p, and neglecting the variation of z: thus r= po os nz sin n - t 9`0/ and therefore r = ro - P cos mz cos (nt- s). 32 Cs Putting ro= a, and writing pa for the corresponding value of po, the approximate equation of the free surface is r = a- - cos mzcos (nt - s0). 33 SC n- - a2 Now by I6 and 26 p = Iio + C2( -- + s cos mz cos (nt - sO) P- = 2 \a2 r2/ and the condition p = H]o gives, with the help of 33, 0 a2 a - -lacosm cos (t -0) 2 J Pa + - cos mz cos (nt - sO); that is, neglecting the squares of small quantities, Z - --- =. 34 a" Also by 29 1 / sc 1 -dw Pa -m drr,=a' thus, with the value of w given in 31, the condition 34 becomes _ sc2 JIs (ma) _ K, (ma) a-) I', (mb) K' (m b) meC2 (ma) K_ (ma) _ a3 I, (mb) K' (mb) 35 XI.] HYDRODYNAMICS. 109 This may be regarded as an equation to find n when the other quantities are given. If we write a= 36 (the angular velocity at the free surface in the steady motion), and _ f (ma) Kr _ (mca)l. 1 (ma) K_ (ma)) \Im ((mb) K' (mib)) + (mb) K (mb)) ' 37 the roots of the equation 35 are given by n = w (s + VN). 38 N is an abstract number, which is positive whenever a, b, m are real and b > a. Thus the steady motion is stable in relation to disturbances of the type here considered. This might have been anticipated, from other considerations. The interpretation of 38 is that corresponding to each set of values m, s there are two oscillations of the type 16, travelling with angular velocities (1 + V) and (I1 - \) s s respectively about the axis of the vortex. A special case worth noticing is when b = co. In this case we must put w = AKs (mr) and 37 reduces to N= - a K (ma) K, (ma) The third case considered by Lord Kelvin is that of a cylindrical core rotating like a solid body and surrounded by liquid which extends to infinity and moves irrotationally, with no slip at the interface between it and the core. Thus if a is the radius of the core, we have U= wr when r <a,} 77wa2 39 U = --- when r > a, for the undisturbed ti. for the undisturbed motion. 110 HYDRODYNAMICS. [XI. For the disturbed motion we start as before with equations 16, and by precisely the same analysis we find w = AJ (cr) when r < a, w = BK8 (mr) when r > a, with KC2= m2 4 2 {4 (n - SW)2} J with K= - ~ — s- — / ' (n - so)2 At the interface p, w and v must have the same value on both sides. Now by 17 and 27 it follows that the values of r are the same when those of w agree; hence the two conditions to be satisfied are, by 23 and 29, A J, (Kca) = BKs (ma), 41 and A (nn - s) {(n- s) KcJ' (Ka) - 2s Js (Ka)} ---,,a= -BK' (ma). 42 m {4W2 ( - (n S)2} Eliminating A/B, we obtain, on reduction, KaJ; (Ka) cK2aK' (ma) 2sw __ J, (ca) r nKs (nma) n - so or, which is the same thing, mKaJ' 2aK (ca) IaK (ma L ) / 2 0, J. ((a) + K (ma) 43 a transcendental equation to find c when the other constants are given. When K is known, n is given by / 2m \ n==w s + I fC2 + m2 For a proof that the equation 43 has an infinite number of real roots, and for a more complete discussion of the three problems in question, the reader is referred to the original paper above cited. Since the expression for z involves the factor sin mz, we may, if we like, suppose that the planes z= 0 and z = r/m are fixed boundaries of the fluid. We will now consider the irrotational wave-motion of homogeneous liquid contained in a cylindrical tank of radius a and depth h. The upper surface is supposed free, and in the plane z = 0 when undisturbed. xi.] HYDRODYNAMICS. 11l The velocity-potential C must satisfy the equation aD2f1D I 2I a =0 ~- + 2 44 Dr2 r ar r r2 2 and also the boundary conditions =0 when z=-h, at \ 45 r= 0 when r=a. ar0 These conditions are all fulfilled if we assume = = AJ( (,cr) sin nO cosh K (z + h) cos mt, 46 provided that K is chosen so that J' (Ka)= 0. 47 If gravity is the only force acting, we have, as the condition for a free surface, -2+g+ =O 48 when z =0, neglecting small quantities of the second order; therefore - m2 cosh Kh + gK sinh ch = 0, or m2 = gfc tanh ch. 49 The equations 46, 47, 49 give a form of 5 corresponding to a normal type of oscillation; when the liquid occupies the whole interior of the tank, n must be a whole number in order that b may be one-valued. The equation 47 has an infinite number of roots K(), Kc(), etc., so that for each value of n we may write, more generally, < = EAsJ( (Kctr) cosh K() (z + h) cos m()t sin n0, 50 and by compounding the solutions which arise from different integral values of n we obtain an expression for > which contains a doubly infinite number of terms. Moreover instead of the single trigonometrical factor A cos mt sin nO in the typical term we may put (A cos mt + B sin mt) sin nO + (C cos mt + D sin mt) cos nO, where A, B, 0, D are arbitrary constants. 112 HYDRODYNAMICS. [XI. As a simple illustration, let us take n = 0, and put ) = JAJo (cr) cosh K (z + h) sin mt; then if, as usual, we write r for the elevation of the free surface at any moment above the mean level, 71 () )= = AJo (/cr) sinh /ch sin mt, 5 and since this vanishes when t = 0, the liquid must be supposed to start from rest. Integrating 51 with regard to t, we have a possible initial form of the free surface defined by = - - A sinh KhJo (Kr), 52 the summation referring to the roots of JO (a) = 0. By the methods of Chap. VI. the solution may be adapted to suit a prescribed form of initial free surface defined by the equation r} =f(r). It will be observed that in 49 Kh is an abstract number; and if, in the special case last considered, we put ca = X, so that X is a root of Jo/(X)= 0, the period of the corresponding oscillation is 27= 2r - coth m V Xg a A specially interesting case occurs when a rigid vertical diaphragm, whose thickness may be neglected, extends from the axis of the tank to its circumference. If the position of the diaphragm is defined by 0= 0, we must have, in addition to the other conditions, =0 when 0=0. 30 This excludes some, but not all, of the normal oscillations which are possible in the absence of the barrier; but besides those which can be retained, we have a new set which are obtained by supposing n = k + i, where k is any integer. Thus in the simplest case, when k = 0, we may put c = AJ (cr)cos c cosh K (z + h) cos mt, 5 3 2 XI.] HYDRODYNAMICS. 113 or, which is the same thing, 0 - =Ar-~ sin (Kr) cos 2 cosh K (z + h) cos nt, with the conditions tan ca - ca = 0, mn2= gc tanh ch '54 The equation tan x - = 0 has an infinite number of real roots, and to each of these corresponds an oscillation of the type represented by 53. More generally, if we put (2k + 1) 7 [ < 7r] n < 77 — ^ ] 2cc where kc is any integer, the function ( = (A cos mt + B sin nt) J, (Kr) cosh K (z + h) sin nO, with the conditions 47 and 49 as before, defines a normal type of oscillation in a tank of depth h bounded by the cylinder r = a and the planes 0 = + a. Similar considerations apply to the vibrations of a circular membrane with one radius fixed, and of a membrane in the shape of a sector of a circle (see Rayleigh's Theory of Sound, I. p. 277). Another instructive problem, due to Lord Kelvin (Phil. Mag. (5) x. (1880), p. 109), may be stated as follows. A circular basin, containing heavy homogeneous liquid, rotates with uniform angular velocity c about the vertical through its centre; it is required to investigate the oscillations of the liquid on the assumptions that the motion of each particle is infinitely nearly horizontal, and only deviates slightly from what it would be if the liquid and basin together rotated like a rigid body; and further that the velocity is always equal for particles in the same vertical. The legitimacy of these assumptions is secured if we suppose that, if a is the radius of the basin, W)2a is small in comparison with g and that the greatest depth of the liquid is small in comparison with a. We shall suppose, for simplicity, that the mean depth is constant, and equal to h. G.. 8 114 HYDRODYNAMICS. [XI. Let the motion be referred to horizontal rectangular axes which meet on the axis of rotation, and are rigidly connected with the basin. Then if u, v are the component velocities, parallel to these axes, of a particle whose coordinates are x, y, the approximate equations of motion are au 1 Dp 2wv = - -. 3~ p 3x ' pav 55 Dav 1 ap -t + 2wu=- ay If h + z is the depth of the liquid in the vertical through the point considered, the equation of continuity is h,r+ aiuv 3v\ = 056 a- + ay at=0; 56 while the condition for a free surface leads to the equations ax ax r ' ap Dz { 57 If we eliminate p from 55 by means of 57 and change to polar coordinates, we obtain au Saz 2a v g - =0, av a 858 where u, v now denote the component velocities along the radius vector and perpendicular to it. The equation of continuity, in the new notation, is D au Dv u\ Dz h -+ r-DO + =0. 59 \or ro0 r / at From the equations 58 we obtain 2 + 4W) u =- grt -gg- 2wg rD '6 /a2, az aZ; 60 /a2 z hence by operating on 59 with ~ + 4a2) and eliminating u, v we xi.] HYDRODYNAMICS. 115 obtain a differential equation in z, which, after reduction, is found to be /2 2 \a 1 a2 ^ +4o,2. ~-g~ Qo+~5+~.- - +. 6 ( at ) at ( 2 K 1 )ZI Let us assume z = cos (mO - nt), where m, n are constants, and ' is a function of r only: then on substitution in 6I we find d2' 1 d. ( 2 d- + r 1- + (cK2 —,) = 0 62 dr2 r dr r2 / 2 -- 42 where K2 -.. 63 gh The work now proceeds as in other similar cases already considered. Thus for instance in the simplest case, that of an open circular pond with a vertical bank, we take m to be a real integer, and put z = J,, (Kr) cos (mO - nt). 64 The boundary condition = 0 when r=a gives, for the determination of n, the equation 2mzo Jm, (Cca) - nca J, (a)= 0. 65 If Wo2 is small in comparison with gh, we have approximately 2 K2 =-y gh and 65 becomes / ^fnay nWa w ( naa 2mo) Jm (r7f) X-. - =( 0. In the general case it will be found that the equations 58 and 60 are satisfied by putting u = Usin (m0-nt), v = Vcos (mn- nt), 66 with U 2 ( dr 2 ) 67r V- = (n-2c_ do r +'). _ ___r (_. 2 r d r' 2 8-2 116 HYDRODYNAMICS. [XI. By assuming for the solution of 62 g = AJan (cr) + BY,, (r) we obtain a value for z which may be adapted to the case of a circular pond with a circular island in the middle. It should be remarked that the problem was suggested to Lord Kelvin by Laplace's dynamical theory of the tides: the solution is applicable to waves in a shallow lake or inland sea, if we put o) = ey sin X, 7 being the earth's angular velocity, and X the latitude of the lake or sea, which is supposed to be of comparatively small dimensions. We will conclude the chapter with a brief account of the application of Bessel functions to the two-dimensional motion of a viscous liquid. It may be shown, as in Basset's Hydrodynamics, II. p. 244, that if we suppose the liquid to be of unit density, and that no forces act, the equations of motion are v2 ap (2 It u 2 v\..- - - rr ) r ar r2' r2 a7 68 68 UV t p _- v 2 tv 6 r raO r r aOl If ~ is the current function, _a8 a ra' V-aor and if we put 8a2*I _@ =, )69 the equations of motion may be written in the form V2 ap 1 a% au, - — = - _ __+ r ar r as at UV ap + a 70 r raO ar at If the squares and products of the velocities are neglected =at' = a and the equations become ap I ax =0) ar r a30 ap a = 7 ao + r -= 86o ar I XI.] HYDRODYNAMICS. 117 Hence, eliminating p, a a 1 a a- _- ( + -1 az _ 0, ar a +g r ao2 or, which is the same thing, VX = 0. 72 A comparatively simple solution may be constructed by supposing that =0, and * = Te2ti, where YP is a function of r only. This leads to d2 1 d mi tdP = 0, drr2 r dr and if we write 2m one value of T is = AJ0o{(l -i) Kr}. We obtain a real function for f by putting = (a + /i) e"tiJo {(1 - i) Kr'} + (a - i) e-mtiJ {( + i) Kr}, 73 a,,, mn being any real constants. Suppose the velocity is prescribed to be ace sin mt when r = a; then aw sin mt = - f) \ar 7= = (1 - i) (a + 3i) etI"J {(1 - i) Ka} + c (1 + i) (a - /i) e-?tiJ {(1 +i) Ka}, and therefore iaco / (1 - i) (a + /3) J (1 - i) ca}- 2 so that ao (1 -i) J,{(1-i)Kr} a w ~ (l+i)J0o (l+i)( cr, e(noi + e t 74 4KfJ,1 l- i) Ka} e9t 4+ 4cJ1(1 + i) Ka e In order to obtain this explicitly in a real form, let us write (I - i) Jo {(1- i) K} + Q K~J, {(1 - i) Ka~ 118 HYDRODYNAMICS. [XI. P and Q being real functions: then Ifr = ao {(P + Qi) (cos mt + i sin mrt) + (P - Qi) (cos mt - i sin mt)} = 1ao (P cos mt - Q sin nt). 75 The boundary condition may be realised by supposing the liquid to fill the interior of an infinite cylinder of radius a, which is constrained to move with angular velocity o sin mt about its axis, carrying with it the particles of liquid which are in contact with it. (This example is taken from the paper set in the Mathematical Tripos, Wednesday afternoon, Jan. 3, 1883.) A very important application of the theory is contained in Stokes's memoir " On the effect of the internal friction of fluids on the motion of pendulums" (Camb. Phil. Trans., vol. IX.): for the details of the investigation the reader should consult the original paper, but we shall endeavour to give an outline of the analysis. The practical problem is that of taking into account the viscosity of the air in considering the small oscillations, under the action of gravity, of a cylindrical pendulum. In order to simplify the analysis, we begin by supposing that we have an infinite cylinder of radius a, surrounded by viscous liquid of density p, also extending to infinity; and we proceed to construct a possible state of two-dimensional motion in which the cylinder moves to and fro along the initial line 0 = 0 in such a way that its velocity V at any instant is expressed by the formula V = ce2nt + coe-2vnot, 76 where v = /p, Ft being the coefficient of viscosity; n, no are conjugate complex constants, and c, Co are conjugate complex constants of small absolute value. The current finction s must vanish at infinity, and satisfy the equation V v a) '=0, 77 and, in addition, the boundary conditions aI= Va cos, -=Vsin, 78 when r a. XI.] HYDRODYNAMICS. 119 Now if we assume =- [e2vnti { + BX (r)} + e-2vnti + Boo (r)f sin 0 79 part of this expression, namely the sum of the first and third terms, satisfies the equation V22 = 0, and the remaining part satisfies (V at) + =0, provided the functions X, X0 are chosen so that drI +-~- 2in +~ X=0, dr2 rf dr + r2 ld2X o ~ dX0 o X den + r r-(2ino+ - X%0. dj,.2 r dr 9"2/ co These equations are satisfied by =K, (1 + i) Vnr, Xo = K (1 - i) Vn0r}. Put (1 + i) Vm = X, (1 - i) Aio = Xo; then X = K, (X i-) = P + iQ,) X=K1(\r)=P-iQ} 80o Xo = K, (Xo,) = P - iN j' where P and Q are real functions of r. The boundary conditions are satisfied if A + BX (a)= ca, -A + B'(a) = c; whence A ca2 {aX ()- X (a) x () )+ x' (a) '8 2ca B f X (a) + ax' (a) and A,, Bo are obtained from these by changing i into -i, 120 HYDRODYNAMICS. [XI. The equations 79, 80, 81 may be regarded as giving the motion of the fluid when the cylinder is constrained to move according to the law expressed by 76. By proceeding as in Basset's Hydrodynamics, II. 280 it may be shown that the resistance to the motion of the cylinder, arising from the viscosity of the surrounding liquid, amounts, per unit length of the cylinder, to Z = 27rTpvnia (Le2^ti - Loe —2loti), 82 where L = —BX) (a) cc{a' (Xa)- 3X (a) 3 (a) + Ca' (a) and L0 is conjugate to L. Let cr be the density of the cylinder: then the force which must act at time t upon each unit length of it, in order to maintain the prescribed motion, is dV F = 7roa- d- + Z = 2rivca2 (Nesvnti - 1oe-2v2oti), 84 with I=V y - { p-aX(a)- 3X (a)l X Ta) + x(a (Ca) J' 85 N0o= the conjugate quantity. Now let us suppose that we have a pendulum consisting of a heavy cylindrical bob suspended by a fine wire and making small oscillations in air under the action of gravity. We shall assume that when the amplitude of the oscillation is sufficiently small, and the period sufficiently great, the motion will be approximately of the same type as that which has just been worked out for an infinite cylinder; so that if 4 is the horizontal displacement of the bob at time t from its mean position, we shall have = V = e2Veti + coe-2v22oti 86 The force arising from gravity which acts upon the bob is, per unit of length, and to the first order of small quantities, -7( -p). a' xI.] HYDRODYNAMICS. 121 where I is the distance of the centre of mass of the pendulum from the point of suspension. Equating this to the value of F given above, we have the conditional equation 2ivl (Ne2vnti - roe-2vnoti) + (o- - p) g~ = 0, 87 which must hold at every instant, and may therefore be differentiated with regard to the time. Doing this, and substituting for t its value in terms of the time, we obtain {- 4nv21N + (a - p) go} e 2"ti + {- 4nov21N0 + (- - p) gc,} e-2noti = 0, which is satisfied identically if we put (- - p) gc = 4nzv21N, or, which is the same thing, ( —p) 4 ( + ax (a) —3X () P) V2. 88 ~I \ (a)+ aX (a) This, with X (a) defined by 80 above, is an equation to find i which must be solved by approximation: since the motion is actually retarded, the proper value of n must have a positive imaginary part. As might be expected, when p is very small in comparison with o, 4v-n2 = g/l approximately. The constants c and Co are determined by the initial values of I and f, together with the equations 86 and 87. CHAPTER XII. STEADY FLOW OF ELECTRICITY OR OF HEAT IN UNIFORM ISOTROPIC MEDIA. CHAPTER VII. above, which deals with Fourier-Bessel Expansions, contains all that is required for the application of Bessel Functions to problems regarding the distribution of potential; but it may be advisable to supplement that theoretical discussion by a few examples fully worked out. We take here a few cases of electric flow of some physical interest. Other problems with notes as to their solution in certain cases will be found in the collection of Examples at the end of the book. In the discussions in this chapter we speak of the flow as electric; but the problems solved may be regarded as problems in the theory of the steady flux of heat or incompressible fluid moving irrotationally, or even of the distribution of potential and force in an electrostatic field. The method of translation is well understood. The potential in the flux of electricity becomes the temperature in the thermal analogue, while the conductivities and strength of source (or sink) involve no change of nomenclature; the potential in the flux theory and that in the electrostatic theory coincide, the sources and sinks in the former become positive and negative charges in the latter, while specific inductive capacity takes the place of conductivity. If V be the potential, then in all the problems here considered the differential equation which holds throughout the medium is a2V -2V a2v + + -, I or in cylindrical coordinates r, 0, z, a2V 1 V 1 a21V V _~ r a + ' '- + - = 0. 2 ar2 r ar y.2 a02 az8 XII.] STEADY FLOW OF ELECTRICITY OR OF HEAT, ETC. 123 At the surface of separation of two media of different conductivities k1, k2 the condition which holds is k,- + kJ, = 0, 3 where?n, n2 denote normals drawn from a point of the surface into the respective media, and V1, V2 are the potentials in the two media infinitely near that point. If one of the media is an insulator, so that say k2 = 0, the equation of condition is a-0. 4 on Let us define a source or sink as a place where electricity is led into or drawn off from the medium, and consider the electricity delivered or drawn off uniformly over a small spherical electrode of perfectly conducting substance (of radius r) buried in the medium at a distance great in comparison with r from any part of the bounding surface. Let it be kept at potential V, and deliver or withdraw a total quantity S per unit of time, then since V = constant/r, V 1 S 4 7rkr' 5 The quantity on the right is half the resistance between a source and a sink thus buried in the medium and kept at a difference of potential 2V. If the electrode is on the surface (supposed of continuous curvature) of the medium the electrode must be considered as a hemisphere, and the resistance will be double the former amount. In this case V^ 1 6 S 27rkr When r is made infinitely small we must have rV finite, and therefore in the two cases just specified Lt rV=4rc 47S& 7 LtrV= 2- k Equations I, 2, 3, and 7 are the conditions to be fulfilled in the problems which we now proceed to give examples of. Those we here choose are taken from a very instructive paper 124 STEADY FLOW OF ELECTRICITY 01R OF HEAT [XII. by Weber ("Ueber Bessel'sche Functionen und ihre Anwendung auf die Theorie der elektrischen Strome," Crelle, Bd. 75, 1873), and are given with only some changes in notation to suit that adopted in the present treatise, and the addition of some explanatory analysis. We shall prove first the following proposition. If V be the potential due to a circular disk of radius r, on which there is a charge of electricity in equilibrium unaffected by the action of electricity external to the disk, then if z be taken along the axis of the disk, and the origin at the centre, V = c e^z sin (Xrl) J (Xr), 8 where the upper sign is to be taken for positive values of z and the lower for negative values, and c is the potential at the disk. In the first place this expression for V satisfies 2; if then we can prove that it reduces to a constant when z = 0, and gives the proper value of the electric density we shall have verified the solution. By 46, p. 18 above, if e > 0, j e-' sin (Xr) JO (\r) x (JO ro A -f e~- sin (-Xr) {f cos (Xr sin 0) do} X dX = 1 d0 e- sin kXr cos (Xr sin 0) a a 9 d e1 9 since changing the order of integration is permissible here. Consider the integral e e- sin (Xr,) cos (Xr sin 0) d. /o This can be written 1 e-ex [sin {\ (r1 + r sin 0)} + sin {X (r, - r sin 0)}] d But we know that if a > 0 fj e. in x= 1 J e-a~ sin xdx= -- Jo a+I XII.] IN UNIFORM ISOTROPIC MEDIA. 125 Multiplying this equation by da and integrating both sides from a = e to a = oo (e > O), we obtain r, ^ Tdx,r je-~x sin x = - -- tan-' e. o 2 Thus the integral considered has the value 7r 1 E 1 e - - - tan-'- -- - - tan2 2 r, + r sin 0 2 - - r sin 0' and fo dx dO e-fe sin (Xr,) cos (Xr sin 0) - 7r2 1 Cr l 1 K -_ _ dO tan- -- dO tan-' I 0 2-2 Jo r+ r sin 0 2 -o -r sin *0 The integral on the left is convergent for all positive values of e including 0. Hence if we evaluate the equivalent expression on the right for a very small positive value of e we shall obtain the value of the integral on the right of 8 when z=0. In doing this there is no difficulty if r > r; but if r < r, the element 7r of the last integral in 0o, for which 0 = sin-l r,/r, is 2 dO, and the integral requires discussion. The first of the two integrals on the right vanishes if e be very small. The second also vanishes when r, > r, so that the integral sought is in that case 7r2/2. Now fdO tan ---1 2 dO ( - tan- r r, - r sin 0 0 2 or2 r i-rsm J 2 - r sin t a-2 tn-' i Ot tan-' dO. ~~~~~~~2 ^ ~ J^~~r Each element of the first integral just written down on the right is -7rdO except just when 0 = sin-' (r/r), when it vanishes, since e is not zero. Similarly except just at the beginning each element of the second is - 17rd0. Hence for r > r, 1 s d0 tan- i r sin- - - 2 J or, - r sin 0 = r 2 126 STEADY FLOW OF ELECTRICITY OR OF HEAT [xn. and we have finally for e = 0 -c sin (Xr,)J, (x) d (\) c, if r < 2c r. II = 2sin-l, if r > rI; 7r r when r, = r, the two results coincide. Thus the expression V = ez sin (Xr) Jo(Xr) 12 7- JO 1 satisfies the differential equation, gives a constant potential at every point of the disk of radius r-, and is, as well as V/az, continuous when z = 0, for all values of r. Lastly to find the distribution, we have for z + 0 4 1 rz 2 27r r/ _ r 3 by I 5 I, p. 73 above. Or the whole density, taking the two faces of the disk together, is C/7r2^/2- r2. This is a result which can be otherwise obtained. Hence the solution is completely verified. We can now convert this result into the solution of a problem in the flow of electricity. Let us suppose that the electrode supplying electricity is the disk we have just imagined, and let it be composed of perfectly conducting material, and be immersed in an unlimited medium of conductivity k. Then to a constant the potential at any point of the electrode is V = - sin (xM,) Jo (xr) 4 7 J ' The sink or sinks may be supposed at a very great distance so that they do not disturb the flow in the neighbourhood of this disk-shaped source. The rate of flow from the disk to the medium is -ea V/az per unit of area at each point of the electrode, and is of course in the direction of the normal. At the edge by 13 the flow will be infinite if the disk is a very thin oblate ellipsoid of revolution, as it is here supposed to be; but in this, and in any actual case, the total flow from the vicinity of the edge can obviously be made XII.] IN UNIFORM ISOTROPIC MEDIA. 127 as small as we please in comparison with the total flow elsewhere by increasing the radius of the disk. The total flow from the disk to the medium is thus S=- 2k f rdr do. Putting in this for aV/az its value we get 4Ck 2, S = o- - r} d = 8ckr,. Thus the amount supplied by each side of the disk per unit time is 4clor, and we have S -- I 5 If the disk is laid on the bounding surface of a conductor the flow will take place only from one face to the conducting mass, and S has only half of its value in the other case. Then S C=7 16 In this case the condition V/nn= 0 holds all over the surface except at the disk-electrode, and of course 2 holds within the conductor. At any point of the disk distant r from the centre aV _2c 1 S 1 az 77Tr - r 2 27kr- \/1 r2 -I We can now find the resistance of the conducting mass between two such conducting electrodes, a source and a sink, placed anywhere on the surface at such a distance apart that the streamlines from or to either of them are not in its neighbourhood disturbed by the position of the other. The whole current up to the disk by which the current enters is S, and we have seen that c is the potential of that disk. For distinction let the potentials of the source and sink disks be denoted by cl, c2; then if R be the resistance between them C 1 -2 R S ' If the wires leading the current up to and away from the electrodes have resistances pl, p2, and have their farther extremities 128 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. (at the generator or battery) at potentials Vf, TV, the falls of potential along the inleading electrode, and along the outgoing are V1 - C1= Spi, c2 - V= Sp2, so that VI- V2-(cI - c2) = S (pl + p2), and R V= - -2-S -(p + p2)}. I Another expression for the resistance can be found as follows. We have seen that the potential at the source-disk is c, also that for conduction from one side of the disk S = 4cerl. For the sink-disk the outward current in like manner is S= - 4c2kr2. Hence S = 2k (carl - cr). But also R = c-c = 2 _-c2 S 2k(cl?1-c2r2)' and cir, = - c2r2, so that _R r, + r2 Rrlr2 = + 19 -4-r r2 4/cr. 4kr 219 From the latter form of the result we infer that 1/4krl is the part of the resistance due to the first disk, 1/4kr2 the part due to the second. This result is of great importance, for it gives a means of calculating an inferior limit to the correction to be made on the resistance of a cylindrical wire in consequence of its being joined to a large mass of metal. From this problem we can proceed to another which is identical with that of Nobili's rings solved first by Riemann. An infinite conductor is bounded by two parallel planes z = + a, and two disk electrodes are applied to these planes, so that their centres lie in the axis of z. It is required to find the potential at each point of the conductor and the resistance between the electrodes. From the distribution of potential the stream-lines can of course be found also. XII.] IN UNIFORM ISOTROPIC MEDIA. 129 The solution must fulfil the following conditions: a2v 1 av a2 V -+ -- + -— =0, for -a < < + a, Dr2 r Dr Dz2 3V =0, for z= +a, r>r,, Dz DV S + = 2r — r —r' for z = + a, r < r1. Z 27r-r, i -r2 According to the last condition the current is supposed to flow along the axis in the direction of z decreasing. The first condition is satisfied by assuming v=f {4 (X) e^z + (X) e-Az Jo (Xr) dX 20 /o where b (X), f (X) are arbitrary functions of X which render the integral convergent and fulfil the other necessary conditions. Without loss of generality V may be supposed zero when z = 0, and hence we must put p ()= - - (X). Thus 20 becomes V = 20 (X) sinh (Xz) Jo (Xr) dX. 2I /o With regard to the other two conditions, by I50, I5I, p. 73, above, f sin (Xri) Jo (Xr) d = 0, when r > r, Jo = - ___-_, when r < rl. Hence if we take 2/ (X) cosh (Xa). X = sin (Xrl), S sin Xr, 1 or -- 22 ' (X) 4=/rkr1 cosh (Xa) X both conditions will be satisfied. The solution of the problem is therefore S f sinh (Xz) dX V 2= 2,hr(- — x-oosh ) sin (Xrl) Jo (Xr) 23 27rkrJo cosh(X9) X G. M. 9 130 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. From this we can easily obtain an approximation to the resistance R between the electrodes. For we have from 23 S r00 A C1 - =2 — rr tanh (Xa) Jo (Xr) sin (Xrl) - 7rkr, 1 S 1 (1- I e-2^X) 1 J o(Xr)sinX A 24 kl-r, I 2e-2xa x 24 If the second term in brackets be neglected in the last expression, we have to a first approximation, since by I I the integral is equal to 7r/2, R - = _ 1 S ~2kr1,' This of course could have been obtained at once from I9 by simply putting r, = r2. To obtain a nearer approximation the expression on the right in 24 may be expanded in powers of r and r. If terms of the order rl/a3 and upwards be neglected, the result is R _ 1 log2 25 2krl 7rkca If the electrodes are extremely small we may put Xr, for sin Xr, and we obtain from 23 S f: sinh (Xz) 26 27kr. cosh (Xa) ) This expression applies to the space between the two planes z= + a. Hence expanding by Fourier's method we obtain sinh (Xz) 2X 1 nr. nrz 27 cosh(Xa) a 1 X + r 2 2a \2al Hence V S. nr. nz f Jo (\r)dX28 7rka i 2 2a J X2+(fo ) 2 8 Now it will be proved, p. 200 below, that if x be positive 2 f sin (x) d Hence putting K2 for (n7r/2a)2 we have f Jo (Xr) XdX 2 f0 d0 fx sin (rX) dX Jo ++ X2 7rJ1 2-1 K + X2 XII.] IN UNIFORM ISOTROPIC MEDIA. 131 But it can be shown that according as:r > or < 0, X sin (rX) dX 7r Jo + X2 -2 29 Thus since 4 and r are both here positive fJo (r)\X dX - e-Ktrd 7e-KrXdX K2 + x ~:w _ 4~-X2 130 Jo I2 f +2 J1 |2^[-1 jI V I2_ Substituting in 28 we obtain 0nr ~rz f: e-Krx\dX S. sin n. z f eKdX V=~ — sn, E-S - 1- 3i 7rka1 2 2a J Vx2I- ' which agrees with the solution of the problem given by Riemann (Werke, p. 58, or Pogg. Ann. Bd. 95, March, 1855). If the conducting mass instead of being infinite be a circular cylinder of axis z and radius c, bounded by non-conducting matter, the problem becomes more complicated. To solve it in this case a part V' must be added to V fulfilling the following conditions: a2V' 1 aV' a2 V' (1) r +- - +-a =0, for r<c, -a<z<+a, * ar2 r ar azx av' (2) z= 0, for z=+a, av' av (3) ar a- = O0, for r= c,-a<z<+a. If we write L2(r) = X - 1 ' M(r)= =- ra( jM2r e-frdX e-XXdX 32 and denote by L, (c), L2 (c), &c. the same quantities with c substituted for r, the conditions stated are found to be fulfilled by S. nr. nz M, (C) L V'= S i sin 2sm L(r). 33 -riet 1 2 2ca ]2(c) For consider the series.qr 37rz b sin + b sin - +....... 34 9-2 132 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. The coefficients b1, b3,..., which are functions of r, can be so taken as to fulfil the differential equation in the medium. Thus the first three conditions are fulfilled. By the general differential equation we have 2b, +l bn 2b, 35 ar2 r ar 35 of which there are two known solutions J e-KrxdX f+e-KrX d J Vx2- J-V- 2 If the first is expanded in powers of r and integrated it is found that it becomes infinite for r =0. We therefore take as the solution of 35 b 3+le-KrAdX where 3n is a constant to be determined by the remaining equation of condition, (3). We obtain in the notation of 32 - S M, (c). nrT 7rTa M 2(c) 23 and the total potential at any point is V+ V'= S sin sin nrz M2(c) L1 (r)- MI(c) L2 (r) 37 V+V'- S sin_-_sin 37 rlca. 2 2a M2 (c) If in 37 we were to put r = 0, z = + a, and could evaluate the integrals we should obtain c - c2, the difference of potential now existing for the given total flow S. This divided by S would give the resistance. When r = 0, L2 (r) = rT, so that the change in the resistance due to the limitation of the flow to the finite cylinder is 2 Ml (c) 2 _e ka n, M2 (c) ka if - be very small. Hence the resistance is approximately C 1 log 2 2 -7 R= - +-e a 38 R=2kr 7rka a e 38 We now pass on to another problem also considered by Weber. A plane metal plate which may be regarded as of infinite extent, is separated from a conductor of relatively smaller conductivity by a thin stratum of slightly conducting material. For example, this may be a film of gas separating an electrode of metal from a con XII.] IN UNIFORM ISOTROPIC MEDIA. 133 ducting liquid as in cases of polarization in cells. We shall calculate the resistance for the case in which the electrode is small and is applied at a point within the conducting mass. Take the axis of z along the line through the point electrode perpendicular to the metal plate, and the origin on the surface of the conductor close to the plate. Thus the point electrode is applied at the point z = a, r = 0. We further suppose that there is a difference of potential w between the surface of the conductor and the metal plate on the other side of the film. This will give a slope of potential through the film of amount w/8 if 8 be the film thickness. If the conductivity of the film be k1 the resistance for unit of area will be 8/k1, and thus the flow per unit of area across the film is wk,/8. This must be equal to the rate at which electricity is conducted up to the surface of the conductor from within, which is k3V/az. Thus if w be the positive difference between the plate and the conductor surface, the condition holds when z = 0,?V - h + w =0, where h = k/kI. Let p, p', be the distances of any point z, r from the electrode and from its image in the surface respectively. Then the differential equation and the other conditions laid down are satisfied by V= S-k -(,)+w, 39 provided that w fulfils the equations -h w = at the surface, and a2w 1 aw a2w -- + =0- 40 a + - a + a-= O, 4 ar' rar aZ ' 4 throughout the conductor. The first term on the right is the solution we should have had if the film had not existed, the second is the increased potential at each point in consequence of the rise in crossing the film from the plate. A value of w which satisfies 40 is given by w= e-AZ= (X) Jo (Xr) d,, 4I o 134 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. where ~ (X) is an arbitrary function of X to be determined. Now since p =(z-a)+ r2, p' =(z+ a)2+ r2, av S z-a S z+a aw az 47rk p3 47wk p' az S a aw 27rk (a2+ r2) + az ' when z = O. Hence the surface condition becomes having regard to 41, hs (a_ +( ( 1 + hX) > (X) Jo (xr) d = 0. 42 27rk (a2+ r2) o But differentiating with respect to a the equation f e-aXJo (r) dA =(a- 1 we get e-a^Jo (Xr) dX = (-. 43 Jo ov (a r2 2 43 This substituted in 42 gives hS Xe-aA 4 (X)= 27k 1 + hx' 27fo 1 + hX Hence w = j, 7 e-h (z+ac) c (Xr)XdX Z+a ro e-x (z+a) Now eh e-(+h) tdt = J a 1+ hX ' so that we have W hS e=- dte- e-ht Jo(Xr) XdX S +a te-tdt B=r e b_ 43 -2whke I r2 ' by 43 zt2+ S S z+a e-tdt 27rkp' 27rkh e h r ^ (^r24 XII.] IN UNIFORM ISOTROPIC MEDIA. 135 by integration by parts. Thus we obtain for the potential at any point z, r rV S (!+1+ ) h e-t edtt ^k p'p) 27 - -h ei Take a new variable ' given by ht = + z + a, and the solution becomes Y (1 1 s Zrr~ch i- e ^ 45 V 47= + p 2tkh o ( +a)2+ r2 The meaning of this solution is that the introduction of the non-conducting film renders the distribution of potential that which would exist for the same total flow S, were there a combination of two equal positive sources of strength S/47rk, at the electrode and its image, with a linear source extending along the axis of z from the image to - oo, and of intensity S _-e h 27vkh per unit of length, at distance g from the point - (z + a). If the conducting mass be of small thickness then nearly enough p = p'= r, and z + a = a. Thus we obtain Y I S e-1 dt ) -J0 _ + 46 27rk (r Jo /ht2+ r26 if as we suppose (z + a)/h may be neglected. If h/r be small we can expand (h2t2 + r2)-2 in ascending powers of t by the binomial theorem and integrate term by term. We thus get h et dt =1 e-t- () 13... (2n-1) (h2 2n i Vh^t2 + r2 r v 1/ 2.4... 2n 1 /h.27 = (- l)n {1.3... (22- 1)}2 47 r r by which the value of the integral may be calculated if r be not too small. Hence if r be very great S h2 27V= 2vTrk r3' 136 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. or the potential at a great distance from the electrode varies inversely as the cube of the distance. We may solve similarly the problem in which the conducting mass is bounded by two parallel infinite planes, the metal plate z= 0, and the plane z= a, and the source is a disk electrode of radius r, with its centre on the axis of z applied to the latter. As before a feebly conducting film is supposed to exist between the metal plate and the conducting substance. We simply add a quantity w, as before, to the distribution of potential which could have existed if there had been no film. Thus by the solution for the infinite stratum with disk electrode worked out above we have S f~ sinh Xz dX S2 i n sh si sin (Xr1) Jo (Xr) X+ w. 48 27r1jo coshXa X The potential w must fulfil the conditions 3w = 0, for z = a (since the flow from the source-electrode is supposed unaffected by w) h — w=O, for z=0, oz besides of course the differential equation for points within the medium. The first condition is satisfied if we take w = 2 cosh X (z - a) 6 (X) Jo (Xr) dX. Also when z = 0, hkaTy W r Sh f sin (Xr,) () d az 227rkrJo cosh (Xa) -2h sinh (Xa) Xq (X) Jo (Xr) dX /o - 2cosh (Xa) (X) Jo (Xr) dx =0. Hence = Sh sin (Xrl) 47rkri cosh (Xa) jcosh (Xa) + hX sinh (Xa) ' and Sh f0 cosh X (z -a) sin (X\r) JO (Xr) dX 27rkr-1 o cosh (Xa) {cosh (Xa) + hX sinh (Xa)} ' XII.] IN UNIFORM ISOTROPIC MEDIA. 137 so that V S j sinh (Xz) + hk cosh (Xz) (. 27rcrlJ cosh (Xa) + hX sinh (Xa) If we denote by Va the potential at the disk electrode we have S sinh (Xa) + hX cosh (Xa) dX V rkrJ o cosh (Xa) + hX sinh (Xa) sin (Xr) Jo(r) X ' and if the area of the electrode be very small S - fsinh (Xa) + hG cosh (Xa) Va = 2rk J cosh (Xa) + h sinh (a) (r) d. This is the difference of potential between the electrodes, that is, the disk and the metal plate. Comparing it with the difference of potential for the same flow through the stratum of the conductor without the plate, that is with half the total difference of potential given by 23 for the two electrodes at distance 2a, which is 2-k0 I tanh (Xa) Jo (Xr) dX, we see that it exceeds the latter by Sh rf XJo (Xr) dx 27rk J cosh (Xa) tosh (Xa) + hk sinh (Xa)} The resistance of the compound stratum now considered is therefore 1 log 2 h 0f XJo (Xr)dX 4kcr 2 ka + 2rk ]Jo cosh (Xa) {cosh (Xa) + hX sinh (Xa)} ' Since the resistance is between the plate and the electrode, which is taken as of very small radius, JO (Xr) = J, (0) nearly, and so we put unity for JO (Xr) in the expansion just found. The last term is the resistance of the film between the plate and the conductor, and in the case of a liquid in a voltaic cell, kept from complete contact with the plate by the disengagement of gas, is the apparent resistance of polarization. Its approximate value, if a/h is capable of being taken as infinitely small, is 1 h I2 log. 27rka a The value of V in 50 can be expanded in a trigonometrical series so as to enable comparisons of the value of V to be made 138 STEADY FLOW OF ELECTRICITY OR OF HEAT [XlI. for different values of r. A suitable form to assume is V= b, cos / -. 53 From this by the equation ha- -w=O 8Z which holds for z = 0, we obtain the condition h cot I = -. * 54 a This transcendental equation has an infinite number of positive roots which are the values of pu. To find the expansion we put sinh (Xz) + hX cosh (Xz) z- a C= C^ Cos fJL ----. cosh (Xa) + hX sinh (Xa) - ' ct a Multiplying both sides by cos {u (z - a)/a} and integrating from 0 to a we obtain from the left-hand side a2X p2 + X2a2' and from the right sin 2 / ha \ 1,a (1+ 2/ )= a1+ a2 h22) by the relation 54. Thus 2a a2 + h2p2 C~ p2 + \a2 a2 + ha + h2/2 Hence S= f Jo() dX = Sa (a2 + h2t2) fJo (Xr) XdX 27/74jJ0o wie (a2 + ha + h p2 + X2a2 -The expansion is thus S a2 + h2pA2 - a er A:4 Cos p -— l lhz —l 55 7kka a2 + ha + h2/2 cos 7V 1 55 by 30 above. The first root of 54 is smaller the greater h is, the second root is always greater than.r, Thus if r be fairly great the first term of the series just written down will suffice for V. Hence for z = a V has a considerable value at a distance from the axis. XII.] IN UNIFORM ISOTROPIC MEDIA. 139 The solution can be modified by a like process to that used above to suit the case of a cylinder of finite radius c. We have to add to V in this case a function V' which fulfils the conditions - = 0, for z=a, h - = V', for z=0, aV aV' + = 0, for r = c, 8r ar and satisfies the general differential equation. The reader may verify that if in the quantities L1 (r),..., L, (c),..., of 32 above, K be replaced by F/a, V+V' S a2 + hh2 M2(c) L (r)-M (c) L2(r) z~-a 7rka a2 + ha + htL2 M2 (c) a As a final and very instructive example of the use of FourierBessel expansions we take the problem of the flow of electricity in a right cylindrical conductor when the electrodes are placed on the same generating line of the cylindrical surface, at equal distances from the middle cross-section of the cylinder. We shall merely sketch the solution, leaving the reader to fill in the details of calculation. The differential equation to be satisfied by the potential in this case is a2v I al 1 a2V a2V - + — ++ =0. 57 Dr2 r ar r2 a02 8 57 If the electrodes be supposed to be small equal rectangular disks, having their sides parallel to generating lines and ends of the cylinder, and the radius be unity, the surface conditions to be satisfied are summed up in the equations 3=0, forz= +a, where (-s < 0< + 0 )=+c for +i3<z<i3+8 t- >z>-(3+8) - 1> z > - (f3 + 3) ( = 0, for all other points. The distances of the centres of the electrodes from the central cross-section are here + (/8 + -1) and the angle subtended at the axis by their breadth is 2q, while the height of the cylinder is 2a. 140 STEADY FLOW OF ELECTRICITY OR OF HEAT [XII. We have first to find an expression for 1' which fulfils these conditions. This can be obtained by Fourier's method and the result is 4cL2001.}) ~ 1 3> = 2+ 2 - sinno cosn 2Y -1 7 I 11[ i o 2m + 1 { 2 /- 2a 210 2 s (2m + 1)7r (2mn + 1) 7r ). (2rm + 1)7r [Cos 2a -cos 2a (3+ 8) } sin 2a '* Now assume A,,,,, A, sin (2m + 1) ) C V= S2Ain '(r) sin - +2a cos nO, 58 m 2a and the differential equation 57 will be satisfied if + (r) be a function of r which satisfies the equation a2t I a 1 2u 2 /2m~ +1 2 a -U + a + +( r it u=0. r2 r+rr -r2 2ae J Hence we put, (r) = J J(i m 1 = () 59 To complete the solution the constant A,,, must be chosen so as to ensure the fulfilment of the surface condition. This clearly is done by writing 4c 1 2 sin no A, n = r2 2m + 1 n' (1) { (2m + 1) 7r (2m+ ) r cos -- 2 / -cos (2 1/ + 8) in which when n = 0, b is to be put instead of 2 sin no/in. To find the effect of making the electrodes very small we substitute 2 sin (2m + 1) r si(2m + 1) r 2sin 2a (O gis) sin - 2a 4a for the cosines in the value Am, n, and Q8 for sin nf sin {(2m + 1) 7r8/4a} n (2m + 1) 7r/4a Remembering that cc8 is finite, and therefore putting 4c68/7r2 = 1, XII.] IN UNIFORM ISOTROPIC MEDIA. 141 we get the solution V r 2t 4 Q (r) /. (2m + 1) r3. (2mn +- 1) 7rz V2 = E E 2n ~' - cos nO sm sin 60 2afo n=o t (1) 2a a where eO =, e =2 = 63=... = =... =2. For an infinitely long cylinder we can obtain the solution by putting in 60 r d, (2m + 1) r -, A, a 2a and replacing summation by integration. Thus we obtain V=2 e cos nO (i sinj (X) sin (Xz) dX, 61 0 Jo 0i\Jn(ix) co as before being 1, and all the others 2. The reader may verify as another example that if the electrodes be applied at the central cross-section at points for which 0 = + a, the potential is given by.mr \ r1 si n a n V = 2r sin n sin n-2 sin n a sin n0 cos 7Z 1 n 1 Ia.r 7T l n 1 1 iJn (i-r)a If a be infinitely small the second part of this expression vanishes and the first term can be written = l g 1 - 2r cos (a + O) + r2 4 1- 2r cos (a- 0) + r2 which agrees with an expression given by Kirchhoff (Pogg. Ann. Bd. 64, 1845) for the potential at any part of a circular disk with a source and a sink in its circumference. The reader may refer to another paper by Weber (Crelle, Bd. 76, 1873) for the solution of some more complicated problems of electric flow, for example a conducting cylinder covered with a coaxial shell of relatively badly conducting fluid, the two electrodes being in the fluid and core respectively; and a cylindrical core covered with a coaxial cylinder of material of conductivity comparable with that of the core. Each of Weber's papers contains a very valuable introductory analysis dealing for the most part with definite integrals involving Bessel Functions. Several of his results are included in the Examples at the end of this volume. CHAPTER XIII. PROPAGATION OF ELECTROMAGNETIC WAVES ALONG WIRES. THE equations of the electromagnetic field were first given by Maxwell in 1865*. They have since been used in a somewhat modified form with great effect by Hertz and by Heaviside in their researches on the propagation of electromagnetic waves. The modification used by these writers is important as showing the reciprocal relation which exists between the electric and the magnetic force, and enables the auxiliary function called the vector-potential to be dispensed with in most investigations of this nature. If P, Q, R, a, a, 7 denote the components of electric and magnetic forces in a medium of conductivity k, electric inductive capacity /c, and magnetic inductive capacity p, the equations referred to are 4^^Kltrw Xy az k+f \P= 1 -8a 8 47r at 4A7r Vy ( a2 (+ K a Q= I a(a aand _aa 1 /R aRQ\ 47r at 4w7r ay - a a 1 /ap aR\ 47r t =-47ra\z - x) aDy I aQ _P\ 47r at 47r ax 8ay * On the Electromagnetic Field, Phil. Traits. 1865, Electricity and Magnetism, Vol. ii. Chap. xx. XIII.] PROPAGATION OF ELECTROMAGNETIC WAVES. 143 From these may be derived the equations ap aQ aR + y =0, 3 ax ay az a a a/3 r_. - + +-+-= 4 ax ay az The first of these expresses that there is no electrification at the point x, y, z, and the second that the magnetic force, being purely inductive, fulfils the solenoidal condition at every point, except of course at the origin of the disturbance. At the surface of separation between two media the normal components of the magnetic induction, and the tangential components of the magnetic force, are continuous. The tangential components of electric force are also continuous. From the equations given above the equations of propagation of an electromagnetic wave can be at once derived. Eliminating Q and R by means of the first of (2), the second and the third of (I) and (4), we get aa at2 47r/~k ~+ ~/ = V^x, 5 and similarly two equations of the same form for 8 and 7. These are the equations of propagation of magnetic force. By a like process we obtain the equations of propagation of electric force &P _a2 at + Kap V 6 &c. &c. Now for the case of propagation with a straight wire as guide in an isotropic medium, the field is symmetrical round the wire at every instant. Therefore there is no component of electric force at right angles to a plane coinciding with the axis. From this it follows by the equations connecting the forces, that the magnetic force at any point in a plane coinciding with the axis is at right angles to that plane. The lines of magnetic force are therefore circles round the wire as axis. Thus we may choose the axis of x as the axis of symmetry, and consider only two components of electric force, one P, parallel to the axis, and another R, from the axis in a plane passing through it. We shall denote the distance of the point considered from the 144 PROPAGATION OF ELECTROMAGNETIC [XIII. origin along the axis by x, and its distance from the axis by p, and shall use for the magnetic force at the same point the symbol H, which will thus correspond to the / of equations I and 2. From I and 2 we get for our special case the equations aP 1 a 47rkP + - (pH), 7 atR p ap aR 3H 47kR + KI - 8 DH aR aP At a ap 9 Eliminating first H and R from these equations we find for the differential equation satisfied by P ap 2p a2P a2p 1 aP 47r/tk - +c - + - — + - o at at2 ax2 p2 p ap Eliminating H and P we see that R must be taken so as to satisfy a slightly different equation, namely, aR a2R a2R 2R 1 aR 1 ai+fc-at2 ap2+pap -p2X I Finally, we easily find in the same way that H satisfies a differential equation precisely the same as II. In dealing with the problem we shall suppose at first that the wire has a certain finite radius, and is surrounded at a distance by a coaxial conducting tube which may be supposed to extend to infinity in the radial direction. There will therefore be three regions of the field to be considered, the wire, the outside conducting tube, and the space between them. The differential equations found above are perfectly general and apply, with proper values of the quantities k, p, Kc, to each region. Taking first the space between the two conductors we shall suppose it filled with a perfectly insulating isotropic substance. The appropriate differential equations are therefore obtained by putting k= 0, in Io and I. If the electric and magnetic forces be simply periodic with respect to x and t, each will be of the form f (P) e(mx-nt) WAVES ALONG WIRES. 145 Let p = ue(m-nt) i i = ve n(mx-nt) i where u, v denote the values of f(p) for these two quantities. Substituting in IO, remembering that k = 0, we find au+ 1 a2 + - - - (m2 - I/C111) u = 0. 12 ap2 app The quantity m2 - Kpn2 is in general complex since mi includes a real factor which gives the alteration of amplitude with distance travelled by the wave along the wire. On the other hand n is essentially real being 27r times the frequency of the vibration. If the wave were not controlled by the wire we should have in the dielectric m2 _- K/Jn2 =0. The velocity of propagation of an electromagnetic disturbance in a medium of capacities Kc, /u is according to theory Jl1/Kcp; and this velocity has, for air at least, been proved to be that of light. If we denote m2 - K/fn2 by p2 and write: for ppi, 12 becomes a2u 1 au1 ~-2 - 1 = 0, 13 which is the differential equation of the Bessel function of zero order Jo (:). In precisely the same way we get from I I the equation aV 1av / 1\ + a + (i V - = 0, I4 the differential equation of the Bessel function of order 1, namely J1 ()An equation of the same form as 14 is obtained in a similar manner for H. Turning now to the conductors we suppose that in them Kc is small in comparison with k. In ordinary conductors c/kl is about 10-17 in order of magnitude, so that we may neglect the displacement currents represented by the second terms on the left in equations I. We thus obtain the proper differential equations by substituting m2 - 4vrklni for p2. We shall denote this by q2 and write the equations G, M, 10 146 PROPAGATION OF ELECTROMAGNETIC [xIII. a2u 1 an 2+- - + I = 0, 15 ad2 + ) V = 0, where X = qpi = pi Vnm2 - 47rktni. Two values of q and 7 will be required, one for the wire and the other for the outer conductor; we shall denote these by q1, Wl, q2, 72, respectively. The general solution of 13 is as we have seen above, p. 11, = aJo ()+ b Yo (), I7 where a and b are arbitrary constants to be determined to suit the conditions of the problem. The solution of 15 has of course the same form, with q7 substituted for:. For very large values of X both Jo (?1) and Y0 (v) become infinite, and we have to choose the arbitrary constants so that u may vanish in the outer conductor when p, and therefore V, is very great. It is clear from the value of Y, (x) given at p. 40 above for large values of the argument, that this condition can only be fulfilled if a = b (7 - log 2), where ry is Euler's constant, it being understood that the real part of the argument is positive. Again, for small values of the argument Y0 becomes very great, so that in the wire we must put b = 0, since r there vanishes. We thus get for the value of P in the three regions, the wire, the dielectric, and the outer conductor, the equations P = A Jo (1) e(mx-nt) i, 18 P = {BJo () + CYo (I)} e(mz-nt)i, 19 P = D {(y - log 2) JO (27) + Yo(1 e (x} -nt) i, 20 in which A, B, C, D are constants to be determined by means of the conditions which hold at the surfaces of separation between the adjacent regions. WAVES ALONG WIRES. 147 From the value of P we can easily obtain the component R at right angles to the axis. Since all the quantities are periodic 8 and 9 may be written in the form (47k - ~ni) R = miH - pniH = miR -. ap Eliminating first H, then R, between these equations we obtain m=-2 - t/Icn2 - 4J<kni p ' H= (4rk - i) 22 qm2 - 4 jFkni - /LIC1np 'D Thus if in the dielectric we put k = 0, and write p2 for m-2 /_KGn2, we get from 19, remembering that ppi= i, R - 8 = {BJoo(f) + CYo(0)} e(mx-nt) i 23 mn 3P P a P23 H Icn aP - __ ten - BJo(4) + CYo(()} e(mx-nt)i. 24 P In the wire on the other hand where qlpi = 7i, [q2 = w2 - 47r/kni], we have R = m P m AJo(i) en(-nt) i 25 = 4 ^rk-4ki aP 47-kli AJo/ (1) e (mx-nt). 26 ql a8l ql Lastly in the outer conductor we have, writing, as at p. 91, Go (X2) for - {( - log 2) Jo (q2)+ Yo (2)}, M^ agp = Mn DG ) e(mx-nt) i 27 q2 aw2 q2 477-7'VaP 47rk% ) H = 4rki a _ 47k2j DG(2) e(mx-nt)i 28 q2 a2 q2 We now introduce the boundary conditions, namely that the tangential electric force and the tangential magnetic force are continuous. From the latter condition it follows that the lines of magnetic force, being circles round the axis of the wire in the dielectric, are so also in the wire and also in the outer conductor. 10-2 148 PROPAGATION OF ELECTROMAGNETIC [xIII. These conditions expressed for the surface of the wire give for p = a BJo ( G) + CYo () AJo (l), Pf {B (P Gyj( 47rrlciA,, 29 and for p = a2 BJo () + CYo)- DGo(q), Kfl {BJ +) ~ Yf( 47rr2iDG (}). 30 BJ{ (M) + Yo'() 2 D G'(q). 3 P q2 Denoting the values of Jo (I), Y (I) for p=a,, and p =a by Jo()i, Yo0()1, Jo(,)2, Y0(:)2, and eliminating the four constants A, B, C, D by means of the equations just written down, we find 47rk1piJof(1) Jo () - fcnq1Jo (?l) Jo (~), 47TrkZpiGo (W2) Jo ()2- 2cnq2Go (w2) Jo ()2 _ 4w7C piJo (1) Yo( )1- cnqlJo (7i) Yo(~), = 3r 47k2piG (72) Yo0()2- - Knq2Go(o2) YO ()2 3' Considering first long waves of low frequency and remembering that Kt is 1/ 2 where V is the velocity of light in the dielectric, we see that p reduces to m nearly, and the real part of m2 is 47r2/X2 where X is the wave-length. Thus if a, is not large pal is very small. Also if a2, the radius of the insulating cylinder, is moderately small, pa2 is also small. Now when al, a2, are small the approximate values of the functions at the cylindrical boundaries are Jo()= 1, Jo(')= 1 Y() log =, Go()=- log e,1 1 Jo~ (I) -- 1. Jo (X) =-1 = G ---- v Using these values for the J and Y functions in equation 31, putting for brevity Sb, X, for the ratios Jo (W)/Jo (7), Go(72)/Go(w), and ca, a2 for 4rk1l, 47rk2, and neglecting terms involving the factors a12a2, aa2 in comparison with others involving the factor aLa2, we find after a little reduction p2 = 2{q( al a(2a2) I 32 al XIII.] WAVES ALONG WIRES. 149 In all cases which occur in practice it may be assumed that q21 is approximately 4wrukzn. Further Kg = n//pV2, so that the last term within the brackets in the preceding expression is i n2 V-1a2 a2 - a2_,2 877r -V2 Vk1k,2 aa2 X. The first of the other two terms within the brackets is V/- 4r-ktzLni 1 -i i4 n /n 47rkak, 4V2 /47r1 a,) and for small values of al, a, the values of b, X, are large. Hence the modulus of the first term, unless the frequency, n/27r, of the vibrations is very great, is large in comparison with that of the third term. The same thing can be proved of the second term and the third. Hence the third term, on the supposition of low frequency and small values of a,, a2, may be neglected in comparison with the first and second. Equation 32 thus reduces to 2= nt 1-i(4/41b i/iUx 1 p 3v lv ^ v ^ 33 H 87r 9k, 6 2 a, a log a '3 og Let now the frequency be so small that q1ac is very small. Then we have Jo(7) qxaii' o (G,).> evq(,ad and X = Go(2) = q2a2i log 2 and it is clear that the second term of 33 bears to the first only a very small ratio unless a2 be very great indeed. In this case then we may neglect the second term in comparison with the first, and we get ni 1 1 y2_ - ___ _. 34 P fV2 22ra2k a2' a, or, since p2 = m2 - K/C=2 = m - n2/ V2, T2 ( 2vrnai log / un" = ~~~~~~a 150 PROPAGATION OF ELECTROMAGNETIC [XIII. The modulus of the second term in the brackets is great in comparison with unity, and hence if we take only the imaginary part of m2 as given by 35 we shall get a value of m, the real part of which is great in comparison with that which we should obtain if we used only the real part, that is we shall make the first approximation to rm which 35 affords. Thus we write instead of 35 n i 1 o =..2 27 36 A V 27' 2 k lal2 log a2 a1 But if r be the resistance of the wire, and c the capacity of the cable, each taken per unit of length, r = l/7rctlo, c = K/(2 log a2/a,) (where Kc is taken in electromagnetic units), and we have 1 +i m == I- /Vnrc, 36' taking the positive sign. This corresponds to a wave travelling with velocity V/2n//Vrc, and having its amplitude damped down to 1/e of its initial amount in travelling a distance 2/2/\/nrc. The other root of m2 would give a wave travelling with the same speed but in the opposite direction, and with increasing amplitude. It is therefore left out of account. We have thus fallen upon the ordinary case of slow signalling along a submarine or telephone cable, in which the electromagnetic induction may be neglected, and the result agrees with that found by a direct solution of this simple case of the general problem. The velocity of phase propagation being proportional to the square root of the frequency of vibration, the higher notes of a piece of music would be transmitted faster than the lower, and the harmony might if the distance were great enough be disturbed from this cause. Further these higher notes are more rapidly damped out with distance travelled than the lower, and hence the relative strengths of the notes of the piece would be altered, the higher notes being weakened relatively to the lower, XIII.] WAVES ALONG WIRES. 151 We can now find the electric and magnetic forces. The electromotive intensity in the wire is given by P = AJo (n) e(m-nt) i, where i = qpi, the suffix denoting that p is less than the radius of the wire. But if the wire is, as we here suppose it to be, very thin, J, (7) = 1, and the value of P is the same over any crosssection of the wire. Hence if 7y denote the total current in the wire at the plane x = 0, when t = 0, we have AJo ()= -/or, and therefore P = 0re(-t) i. 37 Hence realizing we obtain P= o0re -- cos ( -/ -t. 38 / The radial electromotive intensity in the wire is given by 25, which is - r r, so that R = 1-/. V.rce(nx-nt) i 39 Again realizing we find Al 1/nrc 7r\ R = 2yorp ^Jnrce 2 cos 2- t -4 40 R therefore vanishes at the axis of the wire, and the electromotive intensity is there along the axis. Elsewhere R is sensible, and at the surface the ratio of its amplitude to that of P is a, Vn/ic. The magnetic force in the wire is given by the equation [26 above] H = _ 47rAi AJo (W) e(mx-nt)i which by what has gone before reduces to H 2 ope(m-nti 4 I a24 152 PROPAGATION OF ELECTROMAGNETIC [XIII. The realized form of this is 2 -nVrc / nrc H= —2 ope 2 cos / x-nt. 42 By a well-known theorem we ought to have numerically 27rpH= 47r, where y is the total current at any cross-section. For 7 we have here the equation 7y= 70e(m,t) 43 which when compared with 42 obviously fulfils the required relation numerically. Hence the results are so far verified. We shall now calculate the forces in the dielectric. Putting Pop for the electromotive intensity at distance p from the axis in the plane x = 0, and at time t= 0, we find by the approximate values of the functions given at p. 148 above B + C log (ppi)= Po. But at the surface of the wire P, = yor, where r denotes as before the resistance of the wire per unit length. Thus B + C log (pcLi)= ryor. Hence subtracting the former equation we find Pop = 7or -C log a. C can be found from 29 by putting (by I8) A = 70r and eliminating B. Thus we obtain C = 27yo1crV2 very approximately. Hence P = yr ( - 2ic V2 log P em-nt) i 44 of which the realized form is P= or (1-2C1V2logP)e- XOS (/ nr t) 45 - 2Cos x- nt 45 a, 2 XIII.] WAVES ALONG WIRES. 153 From 23, 44, and 36', since p= m nearly, we get R = (1 + i) V270 21 e (x-t) i 46 or retaining only the real part 73 = Z-m>yo / - - I / - 7nt V R 2= V270 -e 2 cos ( 2 x-nt+ 4 47 Finally from 24 and 36 we have H =- 2 0 e (mx-lt, 48 p __ or H= - 2 70 e-$-I c C (/r - nLt). 49 p 2 Thus the solution is completed for slow vibrations in a cable of small radius. So far we have followed with certain modifications the analysis of Prof. J. J. Thomson, as set forth in his Recent Researches in Electricity and Magnetism (the Supplementary Volume to his Edition of Maxwell's Electricity and Magnetism). To that work the reader may refer for details of other applications to Electrical Oscillations. Reference should also be made to Mr Oliver Heaviside's important memoirs on the same subject, Electrical Papers, Vols. I. and II. passim. We shall now obtain an expansion of xJo0()/Jo(x) in ascending powers of x which will be of use in the discussion of the effective resistance and self-inductance in the case of a cable carrying rapidly alternating currents, and which is also useful in other applications when pal, qxa1, q2a2 are not very small. Denoting the function xJo(x)/Jo(x) (or, for brevity, xJ0/Jo) by u, we have by the relations proved at p. 13 above Ja X Jo= -4=-2+x-). 50 ldu 1 Jo J1' Hence - r dx - J, J, udocz x Jo 1 J 2 Jo J1 x J1 Jo 2 m x =-+ -+ -. X X ut 154 PROPAGATION OF ELECTROMAGNETIC [XIII. Therefore du x d x= u (ia + 2)+ x2. 51 Now guided by the value in brackets in 50 we assume X2 u = - 2 +- - + acx4 + a6t +.. Then by 5 x + 4a4x3+6a,6x5+...)x2+( +a4+... -2+ 4 +.... Multiplying these expressions out and equating coefficients we find 4a4 =-2a4 +~-, 6ac = - 2a6 + 2 -4, 8a8 = - 2as + -6 + a42, 10aIo = - 2a6o + Ca + 2ct a 6,.............................. Hence 1 1 1 1 3 a4 25.3 a -6=29 3' a8= 2 ' 32 5' a10= 215.33 5'. Thus we have Jo(x) 2 x2 x4 x6 X8 13x10 x --- ) —~ 3 + + +..+.. 52 JO (X) 4 25. 3 29. 3 29, 2 215 33. + This expansion may be converted into a continued fraction, the successive convergents of which will give the value of the function to any desired degree of accuracy. The result, which may be verified by the reader, is JO (X) 2 X 2 X2 X -2+- 53 Jo () = - 2 4 - 6 - - 53 Using 52 we obtain Jo (iqlal) - ql,2) + q14al4 ql6 l6 J0 (ia, 1)- 2 - +.... qa J/(ijqla,) 4 2. 3 29. 3 Now approximately q2= - 47rFlcni so that J UO(iq a) (47rL,~ (,lna1l2)2 (47rwa1jna 2)4 Jo (iqla,-) l - 25. 3 29-.32.5 + i {4lr/lk-,nal2 (47rtlkna l3 54 +^ -- 4 --- — 4207. 54 XIII.] WAVES ALONG WIRES. 155 The following table of values is given by Prof. J. J. Thomson: 47rr,1 ncLak iqla1Jo (iql a)/Jo (iq al).5 - 2-002 + -124i I - 2-010 + '250i 1-5 - 2-024 + '372i 2 - 2-042 + '50i 2'5 - 2-064 + '62i 3 - 2-090 + -74i This table shows that for values of 4w'rna12k1 up to unity- 2 may still be taken as an approximation to qlJo(W1)/Jowl as above, p. 148. Thus the first term on the right of 33 is the same as before. The last term on the right of 33 in the case of bare overhead wires depends on a large value of a2, and we have to find an approximate value of Go(q2)/Go(q2) for this case. It has been shewn at p. 90, that G,(ix) = iK,,(x); and this relation holds when x is complex. Further, the semiconvergent expansion I42, p. 68, is valid when the real part of x is positive. In the present case, since r2 = iq2a2, we have x = q2a2 = a2 i/4rk722n J - i, which, taking the same sign for each square root, gives x = (1 - i) a2 VJ2tr tn, of which the real part is positive. Now we have iG'(q2) = K,(x), and K (a) = K1(x). Thus we get G o(q2).Ko(x) X Go/(2) Ko'(x) by 142. Hence / X _ _ k2 C - k a' This is small in comparison with the first term unless k2 be very small. Supposing the latter not to be the case, we have the same solution as before. [The approximation stated above for Go(7)/Go(7), and others which hold for large values of the argument of the functions, are important. By the same process as that already used we can 156 PROPAGATION OF ELECTROMAGNETIC [XIII. prove, from the relation Jn,(ix)=iiI,(x) (139, p. 66), and the semiconvergent expansion 143, p. 68, that approximately Jo (_) _ J = ' when 7 = ix, and x is a complex quantity of which the real part is positive, and of which the modulus is large. Again it is worth noticing that, for any value of x with real part positive, we may write by p. 90 Gn (ix) = inK (x) = in Z/7r CO.e i fl cos n7r.e-x 55 by I42, p. 68, if the modulus of x be very large. Similarly G,^(ix) = il-K,' (x) = i-] Kn_ (x)- Kn(x) =i111 / cos (n. - 1) r. e- 55' approximately, if x have a very large modulus. Thus we get Gn(ix) I - 56 G, (ix) - of which Go (q)/G;(r7) - i, is a particular case. In the same circumstances Jn(ix) = innt (X) =i / ex, 57 and further J",(ix) =_-1 -l_,(x)-I()} in- 1 ex {1 n2 + 3 27 —x 8x +' - e ~B 57 = in-1 2 1 e7 when the modulus of x is very great. Thus we have the result Jn(iX) 58 already stated above for n = 0.] XIII.] WAVES ALONG WIRES. 157 As a further illustration the reader may work out the case of oscillations so rapid that both q~a, and q2a2 are very large. Here J = (71) = i Go(72) Jio' XG'(-) so that by 33 n2 / /L b~ 1 V2 qa q2a2 lo ta2 al Thus m2=p + V2 V2+ qa+ q2a2)lo a and approximately = + "=i + %/2rwr (Va/>lc + a2 2 2) 59 The velocity of propagation is thus V, and the distance travelled, while the amplitude is diminishing to the fraction 1/e of its original value, is the product of n by the reciprocal of the coefficient of i within the brackets. The damping in this case is slow, since the imaginary part of m is of much smaller modulus than the real part. Here if a2C212 be small compared with al,2k, as in the case of a cable surrounded by sea water, the outside conductor will mainly control the damping, and nothing will be gained by using copper in preference to an inferior metal. We shall now calculate the current density at different distances from the axis in a wire carrying a simply periodic current, and the effective resistance and self-inductance of a given length I of the conductor. Everything is supposed symmetrical about the axis of the wire. By I8 we have for the axially directed electromotive intensity at a point in the wire distant p (= i/iq) from the axis P = AJo (l) e(mx-nt) i. 60 This multiplied by k1, the conductivity of the wire, gives an expression for the current density parallel to the axis of the wire at distance p from the axis. If the value of P at the surface of the wire be denoted by Pal, Pa, = AJo (q) e(mzx-t) i 6I (p = a,). 158 PROPAGATION OF ELECTROMAGNETIC [xmII. The magnetic force at the surface is E 4 = 7A Jo' (ni) e (mx-nt) i (p = a). Therefore if r be the total current in the wire we have 47rr =- 27raHal, and so r = 7rk-a AJo' () e (mx-nt)i 62 The electromotive intensity P is the resultant parallel to the axis of the impressed and induced electromotive intensities. To solve the problem proposed we must separate the part impressed by subtracting from P the induced part. Now the impressed electromotive force is the same all over any cross-section of the wire at a given instant, and will therefore be determined if we find it for the surface. But since the induced electromotive intensity due to any part of the current is directly proportional to its time-rate of variation, the induced electromotive intensity at the surface must be directly proportional to the time-rate of variation of the whole current in the wire. Hence by 6I if E denote the impressed electromotive intensity E-A'T = Pa, where AT (A' = a constant) is put for the induced intensity parallel to the axis at the surface. Thus E = A Jo( l) - n 27, A'Jo ()} e(nx^-nt) - iqa Jo() ( I3 6 =_ 4rr qal Jo(Cl) _ ) A r. 63 2wk~a12 Jo (1) Putting r for the resistance (= 1/7ra12k) of unit length of the wire and using the expansion above, we get since q'2 =-47r,1klni, I-Li2n2 1 /14n4 E 12 r2 180 r4 +.. iiA 1 /,12 n2 13 + 434 )} in -A'+ 48 r2 8640 r4 r. 64 Or taking the impressed difference of potential V between the two ends of a length 1 of the wire the resistance of which is R we have ( 1 /1,2/Z2 12 1._4_44/14 \ + 12 R2 180 _R4 -in 1 A' (1 1 /X1n2 13 ( -18 4n4 * -zin - IA'+, -2 48^ 2 4 4...) r. 65 2 48 R- 8640 R4 XIII.] WAVES ALONG WIRES. 159 If we denote the series in brackets in the first and second terms respectively by R', L' we get V=R'r+L'T. 66 Thus R' and L' are the effective resistance and self-inductance of the length I of the wire. It remains to determine the constant A'. If there be no displacement current in the dielectric comparable with that in the wire, a supposition sufficiently nearly in accordance with the fact for all practical purposes, and the return current be capable of being regarded as in a highly conducting skin on the outside of the dielectric, so that there is no magnetic force outside, we can find A' in the following manner. The inductive electromotive force per unit length in the conductor at any point is then equal to the rate of variation of the surface integral of magnetic force taken per unit length in the dielectric at that place. Now, if there is no displacement current, H will be in circles round the axis of the wire, and will be inversely as the radius of the circle at any point, since 27rrH= -47rr. Thus if Hr, be the magnetic force at distance r from the axis of the wire Ed.= —4 — AJo-(A e(m.x-nt)i, q~~ r and 4 rdr = - ---- AJo (r) log 2 e(M-nt) 67 I a, a, But this last expression by what has been stated above is A', and r is given by 62. Thus we obtain A'=- 2 log a, and Z'=2llog ~~+ ( - +^ 13 A,n — 68 and log a + 48 R 8640 R4 6 If J,(x Vi) be denoted by X - Yi, aX/3x, Y/ay by X', Y', andx = 2 /ti,n/r, then by 63 we easily find XXY' -X'Y 2 X'+T' R, 69 160 PROPAGATION OF ELECTROMAGNETIC [XIII. L' 21 loga2 +xl XX' + YY L 21log-+.-... 2 70 a1 2n X'2 + Y2 a form in which the values of R' and L' are easily calculated for any given values of x and n from the Table of J (x /i) given at the end of the book. Equation 65 shows the effect of ul on R' and L' at different frequencies. If however the frequency be very great, we must put in 63 J0 (1,)/Jo (1) = i. We find for this case R1' = Vel it nlR, l' =en l -tA'... 7 Thus in the limit R' is indefinitely great, and L' reduces to the constant term - IA'. The current is now insensible except in an infinitely thin stratum at the surface of the wire. The problem of electrical oscillations has been treated somewhat differently by Hertz in his various memoirs written in connection with his very remarkable experimental researches*. He discussed first the propagation, in an unlimited dielectric medium, of electric and magnetic disturbances from a vibrator consisting of two equal plates or balls connected by a straight wire with a spark-gap in the middle, and, secondly, the propagation in the same medium of disturbances generated by such a vibrator guided by a long straight wire. The action of the vibrator simply consisted in a flow of electricity alternately from one plate or ball to the other, set up by an initially impressed difference of potential between the two conductors. Taking the simple case first as an introduction to the second, which we wish to give some account of here, we may take the vibrator as an electric doublet, that is as consisting electrically of two equal and opposite point-charges at an infinitesimal distance apart, and having the line joining them along the axis of x, and the origin midway between them. It is clear in this case that everything is symmetrical about the axis of x, that the electric forces lie in planes through the axis, and that the lines of magnetic force are circles round the wire. The equations of motion are those given on p. 142 above. By * See Hertz's Untersuchungen iiber die Ausbreitung der elektrischen Kraft, J. A. Barth, Leipzig, 1892; or Electric Waves (the English Translation of the same work, by Mr D. E. Jones), Macmillan and Co., London, 1893. WAVES ALONG WIRES. 161 symmetry the component a of magnetic force in the medium is zero, and the equation a+ = o ay az holds, connecting the other two components. This shows that /3dz - ydy is a complete differential of some function of y, z. In Hertz's notation we take this function as a/lat, so that a2HT a~2I ^ —P) 7'=ya^ 72 The equations of motion become then KP a (a2 a(2HI\ lat at \ ay2 aZ,2 aQ a3jj aM _ a3 gt =-ataxaz ' which declare that the quantities a211 a331 a2T a.. Ke P- - + +, cQR + - K y 2 d2 W xy axz' are independent of t. The propagation of waves in the medium therefore will not be affected if we suppose each of these quantities to have the value zero. Thus we assume as the fundamental equations K~P = a2T + a_, =n - a211 axay tcR- - -- - axaz Using these in the equations of magnetic force, 2, we obtain a/a2I I vun)= o, aZ \ at- KL2 I ay k at2 IC which show that the quantity in brackets is a function of x and t G. M. 11 162 PROPAGATION OF ELECTROMAGNETIC [XIII. only. Thus we write ~IH 1 2-_ 1 V21Tn =f (x, t). Jt2 K/ck It is easy to see that we may put f(x, t)= 0 without affecting the electric and magnetic fields, and the equation of propagation is 82Hl 1 at2 73 A solution adapted to the vibrator we have supposed is II =- sin (mr - nt), 74 r where r is the distance of the point considered from the origin, and (< is the maximum moment of the electric doublet. From this solution the electric and magnetic forces are found by differentiation. In cylindrical coordinates x, p, 0 the equation becomes a2II 1 /an a2l ian\ + - + - rap 75 at2 ~~k\aX2 p2 ) 7p since II is independent of 0. Here p2 = y2 + Z2, and hence if we put now P and R for the axial and radial components of electric force, we must in calculating them from II use the formulae p p (ap 76 KP ' 76 KR = - We take the meridian plane as plane of x, z, so that the magnetic force H which is at right angles to the meridian plane is identical with,. Thus a2-S H= -.. 77 The fully worked out results of this solution are very interesting, but, as they do not involve any applications of Bessel functions, we do not consider them in detail. We have referred to them inasmuch as the case of the propagation of waves along a wire, for the solution of which the use of Bessel functions is requisite, may be very instructively compared with this simple case, from which it may be regarded as built up. In the problem of the wire we have II at each point of the medium close to the surface of the conductor a simple WAVES ALONG WIRES. 163 harmonic function of the distance of the point from a chosen origin. We shall suppose that the wire is very thin and lies along the axis of x, and is infinitely extended in at least one way so that there is no reflection to be taken into account. Hence at any point just outside the surface II = A sin (amx - nt + ). If we exclude any damping out of the wave or change of form we see that A cannot involve x or t; it is therefore a function of p. Thus II =f(p) sin (mx - nt + e). 78 Substitution in the differential equation which holds for the medium gives forf the equation a2f 1 af - + (m- - n2Kg)f = 079 ap2 p ap Here n2/m2 is the square of the velocity of propagation. We shall denote m2 - n2K/J by p2 and suppose that p2 is positive, that is that the velocity of propagation is less than that of free propagation in the dielectric. We have therefore instead of 79 azf 1 3f a+- a p2f= O. ap2 P Vp This is satisfied by Jo (ipp) and by Go (ipp) where pp is real. The latter solution only is applicable outside the wire, as f must be zero at infinity. We have therefore in the insulating medium II = 2cCGo (ipp) sin (mx - nt + e), 80 where C and e are constants. Now by 140, p. 67 above, this solution may be written II == 2C cos (pp sinh B) db} sin (mx - nt + e). Putting p sinh b = A, we get n=2G ~-^^pj d5.sin(mz-gt+e), or II = 20 J -- d. sin (mx - nt +). 8 +0 CP2os-2 or s p = jd.sin(mx-1nt1+-e). 11-2 164 PROPAGATION OF ELECTROMAGNETIC WAVES. [XIII. This result may be compared with that obtained above, 74, p. 162, from which it is of course capable of being derived. When ipp is small, we have Go (ipp) =-log 2 so that neglecting the imaginary part we have Go (ipp)=- + log ). Hence at the surface of the wire I =-2 (7 + log) sin (mx - nt + e). 82 If p = 0, that is if the velocity of propagation is that of light, the solution is I = C' log p sin (mx - nt + e), 83 as may easily be verified by solving directly for this particular case. In all cases the wave at any instant in the wire may be divided up into half wave-lengths, such that for each lines of force start out from the wire and return in closed curves which do not intersect, and are symmetrically arranged round the wire. The direction of the force in the curves is reversed for each successive half-wave. When p = 0, the electric force, as may very easily be seen, is normal to the wire, and each curve then consists of a pair of parallel lines, one passing out straight to infinity, the other returning to the wire. CHAPTER XIV. DIFFRACTION OF LIGHT. Case of Symmnetry round an Axis. THE problem here considered is the diffraction produced by a small circular opening in a screen on which falls light propagated in spherical waves from a point source. We take as the axis of symmetry the line drawn from the source to the centre of the opening; and it is required to find the intensity of illumination at any point P of a plane screen parallel to the plane of the opening, and at a fixed distance from the latter. Let the distance of any point of the edge of the orifice from the source be a, and consider the portion of the wave-front of radius a which fills the orifice. If the angular polar distance of an element of this part of the wave-front be 0, and its longitude be 6, the area of the element may be written a2 sin 0d0do. Putting 4 for the distance of this element from the point, P, of the screen at which the illumination is to be found, regarding the element as a secondary source of light, and using the ordinary fundamental formula, we obtain for the disturbance (displacement or velocity of an ether particle) produced at P by this source the expression a sin 0d~d /, -a sn Od sin (m~ - nt), where mi = 27rr/, n = 2rr/T, X and T being the length and period of the wave. Thus, if the angular polar distance f t th e edge of the orifice be 01, the whole disturbance at P is a2r sin 0 sin (m: - nt) dOdbp. Si 0 o Q 166 DIFFRACTION OF LIGHT. [xIv. Let [ be the distance of P from the axis of symmetry, and b the distance of the screen from the nearest point or pole of the spherical wave of radius a, so that the distance of the screen from the element is a (1 - cos 0) + b. Because of the symmetry of the illumination we may suppose without loss of generality that the longitude of the point P is zero. Then the distance: from the element to P is given by = {b + a (1 - cos 0)}2 + (a sin 0 - cos 0)2 + i2 sins2. Since 0 is small this reduces to = b2 + 4a (a + b) sin - 2ag sin cos ~+ '2, or 2a (a + b). 0 ab' _2 =b + b sin2 0 sin 0 cos b +2b. If now we write p for a sin 0, or a0, we have approximately sin2 0 = p2/4a2, so that A= b+ 2-3- - cos ' + 2 -+ 2b b COS 2ab Hence finally if the opening be of so small radius r, and P be so near the axis that we may substitute 1/b for the factor 1/~, we obtain for the total disturbance the expression abX Jo | sin m (b -+ 2b —b cos + 6 p- 2) nt pd ab i 2b b cos 2ab p Separating now those terms of the argument within the large brackets which do not depend upon p from the others, and denoting them by a, so that ==m(b+ gb)- t, we may write the expression in the form A27r rr aX foI yfsin (w + x) pdpdc, abjo Jo or, (C sin +- S cos a), abX XIV.] DIFFRACTION OF LIGHT. 167 where f277 [r 27r/a+5,?,\,2,, 07= cos2- bP b fcos 0 pdpdb,., f271- 2 / r?,\, a,,2 S =| oo sin - b p p cos ( pdpdo. The intensity of illumination at P is thus proportional to a2b2X2 (C2 + S2), and it only remains to calculate the integrals C and S. This can be done by the following process due to Lommel *, depending upon the properties of Bessel Functions. Changing the order of integration in C we have r ff27r 27r fa+b P,\,.), C = c{os 2- (a +b p2 _ p cos \) d pdp. Now considering the inner integral and writing 27r a + b 27r = ^"s% l^ ^ -X- X, X 2ab 2 XbP we have 2cos 2- 2a b P p cos 2d = os c o- x cos () df Jo r = cos, j cos (X cos h) do, since 2,r sin 1' sin (x cos c) do = 0. But r27r rw cos cos (x cos b) do = 2 cos 1j cos (x cos s) d0 o o = 27 cos o Jo. J, (), by 44, p. 18 above. Hence b2X2 j'z C = I2 cos s. xJo (x) dx, where z denotes the value of x when p = r. * Abh. d. k. Bayer. Akad. d. Wissensch. xv. 1886. 168 DIFFRACTION OF LIGHT. LXIV. Similarly we can show that b2X2 [Z so = - 2o sin. xJ (x) dx. 2 These integrals can be expanded in series of Bessel Functions in the following manner. First multiplying I9, p. 13 above, by zx, and rearranging we obtain xnJ,_1 (x) = nX-Jn (x) + x*J/ (x), and hence by integration rx XJ ~Jn_, (x) dx = xJn (x). Integrating by parts and using this result we get cos -!. xJ, (x) dx = cos!^-. xJ1 (x) X a + b2 f, + 2s b ~ sin 1.x2J1(x)d The same process may now be repeated on the integral of the second term on the right and so on. Thus putting ly for the value of ~J when x = z, and writing U = J ()- J3 () +... = (- ) Y J21 (z), u, = (Y) j, () - (jy) J, (Z) +... X (- ) (-I) J2112 Z) we obtain finally, putting 47rr22"-!/b2X2 for z2, - r2 i — Y Us} OS U1 + 2 4 S = w{ — U1- — 2y (14. 5 The values of C and S can thus be found by evaluating the series Ui, US for the given value of z. This can be done easily by the numerical tables of Bessel Functions given at the end of this volume. The series U1, U2 proceed by ascending powers of y/z. Series proceeding by ascending powers of z/y can easily be found by a process similar to that used above. We begin by performing the partial integration first upon cos x2. xdx, and then continuing XIV.] DIFFRACTION OF LIGHT. 169 the process, making use of the equation a (-n Jn (X)) =- X - J+ (X), which is in fact the relation J, (x) = - J () - Jn+ (X), stated in I6, p. 13, above. Thus remembering that X b2 a + b 2 2= 2 2atb x2= wx2, say, we have as the first step in the process C= b2X-.2 Jo (x) cos 2tX2. xdx == W2mProceed ing in this wy we sin ' 7- w- &2 2+.. bX 1.T 1 1 - 2 - i sm~.J0(s)- - J- -z) J1 (z) cos 1uz 11 Ij- J{, (x) cos )tX ) Proceeding in this way we obtain C.+2L y {J( ) )J(z)+...} -sin + i - 2Y, 6 =~r~+ -3 o- qW '6 -2 Z 2 sin 3 where z / x\ 3 /yZmi f* / O ( 2 Csi + sn) J2 V1=,J1(z)-()J3 (z) +. = v (- 1)( Similarly we obtain 2 2 2Y VO — 2Y S = Vrr2 - c os 2y i _ -.i 8 l^ 2 2 2 ^ 170 DIFFRACTION OF LIGHT. [xiv. Comparing 4 and 5 with 6 and 8 we get U1 Cos y + U2 sin y = sin + Vo sin - 1 cos -Y, Z2 U, sin ly - U2 cos y = cos ~ - v cos y - V, sin ly, which give +,= sin (Y+ - U + v, = cos 2 (y + j) { Squaring 4 and 5 and 6 and 8 we obtain equivalent expressions for the intensity of illumination at the point P on the screen; thus if rr2 = 1 _-_(c2 + S2) -2b2X2 (U~ + U2) = a2b y) {1+ V+V - 2 Vo cos(+ y ) a2b2X2 0 y 1 -2V sin y + y- ). 1o The calculation of these U and V functions by means of tables of Bessel Functions will be facilitated by taking advantage of certain properties which they possess. We follow Lorniel in the following short discussion of these properties, adopting however a somewhat different analysis. Consider the more general functions - ( — 1)P Jn+2p (z), I I ~n /^\n+2 = (- 1)P () Jn+2 (), 12 where n may be any positive or negative integer. First of all it is clear that the series are convergent for all values of y and z. Now if in 75 p. 29 above we put x = n = 0, we get 1 -= J. (z) +2J2 (z) + 2J (z) +.... XIV.] DIFFRACTION OF LIGHT. 171 Hence we see that since J0 (z) < 1 each of the other Bessel Functions must be less than 1V/2. It follows that if y/z < 1 the series for U, is more convergent than the geometric series ny\n+2p \z and if z/y < 1, Vn is more convergent than the geometric series /z (n+2p YJ It is therefore more convenient in the former case to use Un, in the latter to use V, for purposes of calculation. If y=z U= V = JO-J+J-... U1 -,= J- J3 + J5 -.. But putting in 42 and 43, p. 18 above, b = 0, we find cos = J (z) -2J (z) + 2J4 ()-.. sin z = 2J1 (z) - 2J, (z) + 2 (z) -... Therefore when z = y Uo = o = {Jo (z) + cos z}, Ul = Vl = I sin z, U2 = V2 = - TJo (z)- cos z}, and generally _ _n p=n-l1 U2n = 2 = (- -) {J (z) + COS Z} - (- 1)+P J2p (Z), = ~~~p=o [ ~. 13 /_-|\n p=nn-l I 2 p=O U28+l= V2n+l = 2 sin z- -s (- l)0nP J2p1 (z) Returning now to I I and 12 we easily find u, + u,1+ = J (z), Vn + Vn+2 = Jn (z) and therefore Zn ( UIl + Un+2) = y2l ( Vn + Vn+). I 5 Also since J_" (z) = (- 1)n J,0 (z), we find, putting - n for n in the second and first of 14 successively, and also z =y UII + U,+2 = (- 1) (V_1 + V_-n+), Vn + Vn+2 = (- 1) 2( U1- + U-n+2). 172 DIFFRACTION OF LIGHT. [XIV. Differentiating I 1, we find a un n an ( + 2 (Yn+2J az z z + y J '() —Y Jn+2 (z)+.. Using in the second line of this result the relation [I6, p. 13 above] J (Z= JL (z) - J-n+l (z) z we get w ~" -z() J ll (z) + ) J () = - ^Un+1. i6 11~ l~ 16 This gives by successive differentiation the equation Un _U m-1 a-2 am-1 azm -- y am-2 Uyf+l - z_ U+l. 7 Similarly we obtain by differentiating 12 and using the relation [19, p. 13 above] Jn (z) = Jn-1 () - Jn (z), ~v, z 0 =- Vn-1- IS8 az y and therefore Vam V mn- l am- z am-1 azn y azm-2 n-1 + yam1 V-1i 19 Again differentiating the first of 9, we get aLT, av, V z / 2\ -+ -a- = - os (Y + az 4z y y y But by I6 and 18 this becomes - 2+ V= cos=O +-). Differentiating again we obtain U+ V _ =-sin Y+, az z y or by I6 and 18 U + V_1 = - i +. XIV.] DIFFRACTION OF LIGHT. 173 By repeating this process it is clear that we shall obtain 2n+= (- sin ( + 2) = (- sin + 2 20 U2n+2 + V-2n = (- 1)n cos + j If in these equations we put n = 0, we fall back upon 9. Putting in 9 the values of the functions as given in the defining equations I I and 12 we obtain the theorems i/\2+ /2\ Z\ 2p+1' / 2\ - (1)P +() } JP+ =(Z)-i y +s) I (- )P ( + (Z) }J2P+2 (Z) = Jo (z) - Cos 2Y + which include the equations sin z== 2J1 () - 2J (z) + 2J (z) -... cos z = J (z)- 2J2 (z) 2J4 (z)-... as particular cases, those namely, for which y = z. By Taylor's theorem we have a h7 T 7 22 aU U(y, z+ h)= U, + h - - +**** Calculating the successive differential coefficients by means of I6, and rearranging the terms we obtain TT ( ~X t,\ n O ) IhlAS2 +A)2z Un (y, z + h) = Un -- (2z + ) h (2z + h)2 uni 2! (2y)2 f+2 = c (- 1)P h (2z h) Un+p. 22 P [ (2y)P Similarly we can prove that Vn (y, z + h) hp (2z + h)pP (y, z + h) = S -T --- V 23 p! (2y)P These expansions are highly convergent and permit of easy calculation of U, (y, z + h), Vn (y, z + h). The functions Un+, Un+2, Un+3,..., Vn-_, V,_2,..., can be found from Un, Vn, by using I6 and 18 to calculate Un+, Vn-_, and then deducing the others by successive applications of 14. Differentiating I I and 12 with respect to y, and using in the resulting expressions the relation 20, p. 13 above, namely, flJn (z) =,Z Jn,- (z) + z J^rl+ (z), 174 DIFFRACTION OF LIGHT. [xiv. we find 2U =Yi a - 2U}, 1 12-+1 =-Y2 a -y2 Uvn-lJ 24 IZ, = _ y2 Y2 nfl ay Now if u be a function of y we have am (y 2U) am-2u a+ m- a-1 amGu ay I ayM- aym-l aym a =(rn-i)?m-+2my +Y2^ Using this theorem we find by successive differentiation of 24 am Un+ 2 (am+ Un am Un-1 2 aym - aym-+ 2 aym + 2y m U2 a 1 Am- + (an am-T 2 aym-2 -) 2 ay Y k aym~+l 2aym } dz az + ay dz If y = cz dz 1 c Un,,_- Un)+, by I6 and 24 above. By successive differentiation, and application of this result we obtain d2 n= (c2Un2,U, 1 2 U n+i), d2 a^ ay m < 41' C2 a and generally dmUn 1 m (m -1 )... (m - p' 1) m-2 m +p. 2 dzl"~ = 2"' d (- 1)' P8y dz XIV.] DIFFRACTION OF LIGHT. 175 Similarly it can be shown that dm1V7, 1 m (- 1)...( - p l)c+28 dzm 2m ( -p n-1m+p 28 The calculation of the differential coefficients can be carried out by these formulae with the assistance of 14 which now become Un + UT,+2 = C Jn (z), Vn + Vn+2= zJn) 29 We conclude this analytical discussion with some theorems in which definite integrals involving Bessel Functions are expressed in terms of the U and V functions. By I above we have b27o2 fZ C= 27r I cos#. XJ(x)dx. Now let x = zu, then X2 1 = I y = IYU2, therefore since z2 = 47r2/2r2/X2b2 r1 C= 27rr2 cos (yu) uJ, (zy) du. 30 Jo Similarly we obtain S = 27rr2 f sin (yu2). uJo (zu) du. 31 J But equations 4 and 5 give 7r92 Ccos y+Ssin y= -U1, 7rr2 C sin y - Scos y - U2, and these by 30 and 31 give the equations ( JO (Zu) * cos y (1 - 2u2) udu = - U1 r0l~ \~ Ut ^32 J (ZU). sin ~y (1 -u2). udu = UJ ^o y } 176 DIFFRACTION OF LIGHT. [XIV. Differentiating with respect to z we get, since J; () = - J1 (z), j (zu) cos y (1 - u2). 2du = - - z = 2 U2; and similarly nl z J1 (zu) sin Iy (1 - )t2). u2du = U3. Now if we assume f Jn- (Z). cos y (1 - 2). ndu () Un 33 Jo y \y/ and differentiate, making use of the relation Jn-1 (z6) = 1 — J J (zu) - J (zi), we easily obtain Jn (zu) cos (1 - U2). n+du+l =d -() Un+l ~ 2 Y Thus if the theorem 33 hold for any integral value of n it holds for n + 1. But as we have seen above it holds for n = 1, it therefore holds for all integral values of n. Similarly we obtain n1 I ^n-2 J,_2 (za) sin 2y (1 - u2) ue-lddu = - () U 34 Jo Y Y/ The values of C in 6 and 30 give zs sin l/ COS oz Y 2y Y Y A COS 2y^2. Z6Jo (Z%6 d= - sin 2sinsn- VO _ cos Fi. 35 Similarly those of S in 8 and 31 give f sin yu2. uJo (zu) du = c 1 z-cs VO- 36 2o 7 Y 2y 3y y If instead of u we use the variable p (== ur) where r is the radius of the orifice, and write y = kr2, z = Ir, we have instead of 35, 36 ri' j~~1 12 1 1 p Jo ( p) cos (kp2). pdp = sin + sin (kr2) Vo - cos (kr2) VT, Jo () s1 12 1 1 oJ (ip) sin (/kp2). pdp - COS - - - cos (r) V - sin (kr2) DIFFRACTION OF LIGHT. 177 If now r be made infinite while I and k do not vanish, Vo and V1 vanish, and we have rocI 12 JO (1p) cos (Ep2). pdp = sin 2 1 ' 37 JO (1p) sin (kp2). pp =, cos J formulae which will be found useful in what follows. They are special cases of more general theorems which can easily be obtained by successive differentiation. We come now to the application of these results to the problem stated above. Of this problem there are two cases which may be distinguished, (1) that in which y = 0, (2) that in which y does not vanish. The first case is that of Fraunhofer's diffraction phenomena, and has received much attention. We shall consider it specially here, and afterwards pass on to the more general case (2). When y=O, either a=oo and b = o, or a=-b. In the former case the wave incident on the orifice is plane, and the parallel screen on which the light from the orifice falls is at a very great distance from the orifice. This arrangement, as Lommel points out, is realised when the interference phenomena are observed with a spectrometer, the telescope and collimator of which are adjusted for parallel rays. The orifice is placed between the collimator and the telescope at right angles to the parallel beam produced by the former. When a= - b, a may be either positive or negative. When a is negative the orifice is to be supposed illuminated by light converging to the point-source, and the screen is there situated with its plane at right angles to the axis of symmetry. This can be realised at once by producing a converging beam of light by means of a convex lens, and then introducing the orifice between the lens and the screen, which now coincides with the focal plane of the lens. When a is positive, and therefore b negative, the light-wave falls on the orifice, with its front convex towards the direction of propagation. The interference is then to be considered as produced on a screen passing through the source, and at right angles to the line joining the source with the centre of the orifice. G.M. 12 178 DIFFRACTION OF LIGHT. [XIV. This case can be virtually realised by receiving the light from the opening by an eye focused on the source. The diffraction pattern is then produced on the retina. Or, a convex lens may be placed at a greater distance from the source than the principal focal distance of the lens, so as to receive the light after having passed the orifice, and the screen in the focal plane of the lens. The screen may be examined by the naked eye or through a magnifying lens. If a magnifying lens is used the arrangement is equivalent to a telescope focused upon the point-source, with the opening in front of the object-glass. This is Fraunhofer's arrangement; and we shall obtain the theory of the phenomena observed by him if we put y = 0 in the above theoretical investigation. Putting y= 0 in 3, we have 2 2 2 - U= - J(z), U2 = 0, so that, writing M2 for C2 + S2, we obtain M2- J (z)} 38 Airy gave* for the same quantity the expression, in the present notation, z2 z24 6 )2 2.4 ' 2.4.4.6 2.4.4.6.6.8+ which is simply the quantity on the right of 38. By means of Tables of Bessel Functions the value of M can be found with the greatest ease, by simply doubling the value of J1 for any given argument, and dividing the result by the argument. The result is shown graphically in the adjoining diagram. The maxima of light intensity are at those points for which J, (z)/z is a maximum or a minimum. The minima are those points for which J1 (z)= 0. Now when J, (z)/z is a maximum or a minimum z Ji ( == 0. X_ * Camb. Phil. Trans. p. 283, 1834. XIV.] DIFFRACTION OF LIGHT. 179 But 1 j(z) _ (z) _1 J2 () 0 Z2 Z so that the condition becomes J2 (z)= 0, which (20, p. 13 above) is equivalent to 2J (z-J - (z) = 0. Z 15 14 12 1 1,o 9 - -- -- ---- - -2 1000 M 7 6 5 7 -- -- -- --- --- -- -- -- -- -- -- -- -- -- -- - 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Thus when the maxima and minima values of 2J1 (z)/z have been calculated, their accuracy may be checked by observing whether they are also the values of J0 (z) given in the Table for the same arguments. Or the arguments, which are the roots of J2(z) =0, having been obtained, the corresponding values of 2Jf (z)/z are given by the Table of Jo (z) either directly or by interpolation. Places of maximum intensity alternate with places at which the intensity is zero, the light being supposed of definite wavelength, and therefore monochromatic. The roots of J (z) = 0, which are the values of z for which the intensity is zero, can be calculated by the formula given at p. 49 above (Chap. V.). It is there shown that the approximate value of the large roots of J1 (z)= 0, 12-2 180 DIFFRACTION OF LIGHT. [XIV. is (m + ~) wr, and of Ja (z) = 0, is (m + ~) 7r, where m is the number of the root. Hence for great values of z the difference between the values of z for successive maxima or minima is approximately 7r, and the difference for a zero and the next following maximum is Irr. The rings are thus ultimately equidistant. The difference of path of the rays from opposite extremities of a diameter of the orifice to the point P is 2r tan-l c/b, that is 2r'/b or Xz/7r. The distance in wave-lengths is therefore z/Tr. The following Table gives the values of z corresponding to maximum and zero values of 2z-J, (z), which are contained in col. 2. Col. 3 contains the corresponding values of M2. z (roots of J2 (z) =0) 2z-J, (z) M2 0 3-831706 5-135630 7-015587 8-417236 10-173467 11-619857 131323690 14-795938 16'470631 17-959820 19'615861 +1 0 - 0132279 0 + 0-064482 0 - 0-040008 0 + 0027919 0 - 0'020905 0 1 0 0-017498 0 0-004158 0 0-001601 0 0-000779 0 0-000437 0 For large values of z (p. 40 above) is available. and more approximately the semi-convergent expansion of J, (z) As z increases this expansion gives more 2J 2 /2. J1 (z) = sin (z - r), and therefore 2 = -sin2 (- 7r). As the value of z approaches (m + i) wr that of sin2 (z - ~-r) approaches 1, and so the ultimate value of M2z3 is 8/7r. XIV.] DIFFRACTION OF LIGHT. 181 The whole light received within a circle of radius z is proportional to f Mzdz = 4 z-1J (z) dz. But J~ )- = {Jo () -J' (,)} J (Z) = -Jo (z) Jo (z) - J, (z) J (). Hence r Mzdz 2 {1 - J()- J 2(z)}. 39 J If z is made infinite the expression in the brackets becomes 1. Hence, as has been pointed out by Lord Rayleigh*, the fraction of the total illumination outside any value of z is J0 (z) + J (z). But at a dark ring J1 (z) = 0, so that the fraction of the whole light outside any dark ring is J' (z). The values of this fraction for the successive roots of J1 (z) = 0, are approximately '161, '090, *062, ~047,..., so that more than - of the whole light is received within the second dark ring. In the more general case of diffraction, contemplated by Fresnel, y is not zero, and we have M2 = (U2 + U2), 40 and U1, U2 can be calculated by the formulae given above from the Tables of Bessel Functions at the end of the present volume, equation 3 being used if z > y, and 9, with the expansions of Vo and V, if z < y. The maximum and minimum values of M2 are given in the Table below for the values of y stated. We also give here some diagrams showing the forms of the intensity curve for the same values of y. The curves are drawn with values of z as abscissae, and of M2 as ordinates. * Phil. Mag., March, 1881; or 'Wave Theory of Light,' Encyc. Brit., 9th Edition, p. 433. 182 DIFFRACTION OF LIGHT. y = 7r. [XIv. z 3-831706 4715350 7-015587 8-306007 10-173467 11-578479 2 - U' y - 0-122609 - 0'178789 + 0-013239 + 0'074093 - 0-002313 - 0-043104 2 y + 0106159 0 - 0-040631 0 + 0016225 0 AM2 0-026305 0-031966 0-001826 0-005490 0-000269 0-001858 Min. Max. Min. Max. Min. Max. ICI t tiI tt I12- - 10 - -5 8 _ — - - -- - - - -— O- M 0 1 4 5 6 7 8 9 10 11 12 ~~~1 --- z 3'030827 3-6255773 3-831706 7'015587 9-440724 10-173467 2 - U y + 0114161 + 0114593 + 0'114492 - 0-002099 - 0-134688 - 0118330 y= 5r. 2 - U2 y 0 0 + 0'004496 + 0173617 0 - 0067421 M2 0-013033 Min. 0'013132 Max. 00-13128 Min. 0'030147 Max. 0-018141 Min. 0-018548 Max. -I I I I I 1 1 I -I Yltrl I I \ I I 1PM2 ' / l1 \ 1 I I I 0 I 2 3 4 5 6 7 8 9 10 11 12 XIV.] z 2-649454 3-831706 4431978 7-015587 10-173467 DIFFRACTION OF LIGHT. 183 2 U y + 0-067178 + 0'068485 + 0-068964 + 0045384 - 0017711 y = 97r. 2 2 U2 0 - 0010782 0 + 0076624 + 0'048204 0-004513 0-004806 0-004756 0-007931 0-002637 Min. Max. Min. Max. Min. 9 2 7 5 - i -- -- -8 -- -- -- - - I- -- 0 1 2 3 4 5 6 7 8 9 10 11 12 The maximum and minimum values of M2 are those for which aM2 = 0. az But =2 /2\ 2u, TYau2 2= 2 (2 (-oU( + U2a aZz \y) az az _-(2)2zU2(2+U) =- 2() J () 2. 41 Hence a maximum or a minimum is obtained when either J1 (Z) = - J) = o0 ^ -y1- 4~2 au = = - aI =0 ' 4 U -z az II or Thus a value of z which gives a maximum or minimum of J, (z) or of U1 (z) gives either a maximum or minimum of M2. The roots of J, (z) = 0 are given at the end of this treatise, and 184 DIFFRACTION OF LIGHT. [xiv. are values of z which give a maximum or minimum of illumination. The values of 2 Ul/y, 2 U2/y which correspond to these values of z, are obtained by interpolation from those of UC, U2 or V0, V1 for the values of z for which Tables of Bessel Functions are available. The formulae of interpolation are 22, 23 above. The maxima and minima which arise through the vanishing of U2 are found in a similar manner. Supposing it is required to find the roots of U,, the tabular value of U, (z) nearest to a zero value is taken, and the value of z + h which causes U2 to vanish is found by means of the expression on the right of 22 equated to zero, with 2 put for n, that is from the equation O()-, h (2z + h) +. I 2 (2 + h)2, U ( ) 1 t2y Ua (z) + 2-] (2y)2 U4(Z)-... = 0. 43 2 y 2! (2 y)2 Since the series is very convergent only a few terms need be retained; and the value of h (2z + h)/2y found, and therefore that of h. Values of z which render U2 = 0, being thus found, those of 2Ul/y for the same arguments are calculated. The squares of these are the values of M2 which correspond to the roots of U2 = 0. Elaborate Tables, each accompanied by a graphical representation of the results, are given by Lommel in his memoir. The short Tables with the illustrative diagrams given above will serve as a specimen. We conclude the discussion of this case of diffraction with an account of an interesting graphical method of finding, for different values of y, the values of z which give maxima or minima. This is shown in the next diagram. The axis of ordinates is that of y, and the axis of abscissae that of z. Lines parallel to the axis of y are drawn for the values of z which satisfy J1 (z) = 0. These are called the lines J1 (z)= 0. On the same diagram are drawn the curves U,/y2=0. These are transcendental curves having double points on the axis z= 0, as will be seen from the short discussion below. Let now the edge of a sheet of paper be kept parallel to the axis of z and be moved along the diagram from bottom to top. It will intersect all the curves. The distances from the axis of y along the edge of the paper in any of its positions to the points of intersection are values of z, for the value of y for that position, XIV.] DIFFRACTION OF LIGHT. 185 which satisfy 4; and are therefore values of z which with that value of y give maximum or minimum values of M2. 71 or 30 ---- -- 27- -4 26- -- - o~r / xL 21 24 -20 -19 18 37r 12 7 271 4.... 7r/ 0 I 2 3 4 5 6 7 8 9 I0 11 IU 4 I l The equation of the curve U = 0 differentiated gives a2 au2dy a + ay d = az ay dz that is, since by I6 and 24 above, a u2/az = - Ug. Z/y, a u2/ay = UI + - U3. (z/y)2; dy _ 2 z U3 ^ y 3 44 186 DIFFRACTION OF LIGHT. [XIV. When z=O, that is where the curve meets the axis of y, V0= 1, V1 = 0, so that by 9, Z, = sin ly, U2 = 1 - cos ~y. Thus U2 = 0, whenever cosy = 1, that is when ~y = 2m7r, or y = 4m7r. The curve U2 = 0 therefore meets the axis of y at every multiple of 47r. But this value of y makes U1 and likewise U3 Zl/y {= J1(z)- U1 z/ly} zero, so that 8U2 o a U o = o, 0= o. az ' ay The value of dy/dz is therefore indeterminate at the points on the axis of y, that is each point in which the curve meets the axis of y is a double point. If y', z' be current coordinates the equation of the pair of tangents at a double point is Z2 2+ 2z' (y' - y) + (y )2 = 0. 45 It is very easy to verify by differentiation and use of the properties of the functions that when z = 0 and y = 4mnr, a2u 1__ a2U2_o U _ _ 2- = 0 22 -~" =2- ' yz a " - so that the equation of the tangents reduces to (y' - 4m7r)2 = 2z2. Thus the equations of the tangents are y'- 4m7r = 1~2z, y'- 4m7r = - /2z', and their inclinations to the axis of z are given by tan ~b = ~ J2, that is = + 540 44' 8"2. Where the curve meets the axis of z we may regard U2 = 0 as equivalent to the two equations y2 = 0, U/y2 = O, so that the curve U2= 0 splits into two straight lines coincident with the axis of z, and the curve represented by U2 =0. Y2 XIV.] DIFFRACTION OF LIGHT. 187 We have by 3, yU2)- 2 J4(z )+4 J()-.... Hence for the last curve dz a / 2 o when y =0. Thus the branches of the curve U2/y2=0 cut the axis of abscissae at right angles, as shown in the diagram on p. 185 above. We shall see below that this intersection takes place at points satisfying the equation J (z) = 0. Thus the curves U/y2, when y = 0, touch the curves J2 (z) = 0. It will be seen from the curves in the diagram that the value of dy/dz is negative so long at least as y < z, that is, as we shall see, within the region of the curve corresponding to the geometrical shadow. But at points along a line J,(z)= 0, 2_ dy_ y dz '0 2 and is positive so long as z > y, that is also within the region of the geometrical shadow. Hence no intersection of a line J(z) = 0 with the other curves can exist in the region of the diagram corresponding to the geometrical shadow. To settle where the maxima and minima are we have to calculate 2M2/az2. Now a2M2 (2 2a ~x2 =-2 )y 2 (Jy z 2) =2 (-) { JU2-J 2 + J, (J1- U1} 46 Thus considering first points upon the lines J, = 0, we have a maximum or a minimum according as J U2 is positive or negative. On the other hand, when U2 = 0 the points on the curves U2 = 0 are maxima or minima according as J1 U z/y {= J, (J - Ulz/y)} is negative or positive, or as J < or >- UJ1.y 188 DIFFRACTION OF LIGHT. [XIv. Calculating a3M2/a23 we see that this does not vanish for points satisfying the equations J1 (z) = 0, U2= 0, that is wherever a line J1 (z) = 0 and a curve U2 = 0, intersect there is a point of inflexion of the curve of intensity, drawn with M2 as ordinates and values of z as abscisse. It follows by the statement above as to the inclination of the curve within the region corresponding to the geometrical shadow, that within that region there can be no point of inflexion on the intensity-curve. Also, as can easily be verified, there are points of inflexion of the intensity-curve, wherever the curve U2/y2 = 0 has a maximum or minimum ordinate. Referring now to the diagram, p. 185, we can see how to indicate the points where there are maxima and minima. For pass along a line J = 0 until a branch of the curve U2/y2 = 0 is crossed. Here clearly U, changes sign, while J, (z) does not. Thus Jo(z) U2 changes sign, and so all points of a portion of a curve 1J (z)= 0, intercepted between branches of the other curve, give maxima, or give minima, according to the number of branches of the latter which have been crossed to reach that portion by proceeding along J,(z) = 0 from the axis of z. J, (z) U2 is negative for the first portion, positive for the second, and so on, the number of crossings being 0, 1, 2,.... If we pass along a curve U2/y2 = 0 and cross J, (z) = 0, then J,(z) changes sign, but not so U,; for by 14 when J,(z)= 0, U3 = - U, and U, is a maximum or a minimum, since U = 0. But it must be further noticed that when for a branch of the curve U2/y2 = 0 the value of dy/dz is zero, that is when U3z/y = 0, Us changes sign while J, (z) does not; also for U3 = 0, Ja (z) = 0 U,m and because of U2 = O, a U1/az = 0, so that U,1 is a maximum or a minimum. At these points therefore a2M2/az2 changes sign, and hence they also separate regions of the curve U2/y2 = 0 which give maxima from those which give minima, when the process described above of using the diagram is carried out. The first three successive differential coefficients of M2 all vanish when z= 0, and y= 4m7r, that is at the double points, XIV.] DIFFRACTION OF LIGHT. 189 and as there, as the reader may verify, a4M2 3 1 aZ~ 2 ~z2r2 the double points are places of minimum (zero) value of M2. The regions of the curves U/y2 = 0 can now, starting from the double points, be easily identified as regions which give maxima or minima when the diagram is used in the manner described above. To mark regions which give minima they are ruled heavy in the diagram; the other regions, which give maxima, are ruled light. Thus the first regions from the double points to a maximum or minimum of the curve, or to a point of crossing of J (z)=0, whichever comes first, are ruled heavy, then the region from that point to the next point at which U, changes sign is ruled light, and so on. The lower regions of the curves J1 (z)= 0, from the axis of z to the points of meeting with U/y2 = 0, are ruled heavy; the next regions, from the first points of crossing to the second, light, and so on alternately. Thus the whole diagram is filled in. As we have seen U = Vo-cos ( y+ )=Jo()-(- 2J2(Z)+...-cos( y+ ). Hence as y increases in comparison with z, the equation U2 = J0 (z) - cos 2 more and more nearly holds. The reader may verify that the curve J0 (z)- cos ~y = 0 meets the axis of y at the same points as the exact curve, and has there the same double tangents. On the other hand, if y be made smaller in comparison with z, then by 3 we have more and more nearly 2 2 2 2 U=- J(z), U,=-2Y 2(z), so that the branches of U/y2 = 0 approach more and more nearly to the lines J2(z)=0. Thus we verify the statement made at p. 187 above. The value of M2, namely (-) (U U+ U2), 190 DIFFRACTION OF LIGHT. [XIV. with increasing z and stationary y, that is with increasing obliquity of the rays, approaches zero. Hence at a great distance from the geometrical image of the orifice the illumination is practically zero. Consider a line drawn in the diagram to fulfil the equation y= cz. A line making the same angle with the axis of y would have the equation y - z. Let us consider the intensities for points on these two lines. Since y/z=c for the first line, then for any point on that line U1 = cJI - cJ3 +... L / 1 I - 13 1 =sin (c+ - J-CJ3 +C-... U2 = c2J2 - 4J4 +... =-COS 2z ( + C).+Jo- C2 J2 + 4 4 47 For the other line we have, accenting the functions for distinction, U1 1 1 1 L CJ1 C3 J3 + ** C C3 = sin \{z (c + - (cJ, - C3J + c5J -...),; -1 1 -2 = -J2 -..J4 +... = - COs (+ ) 2 C2 2 47' U' Therefore UC + U = sin { (c + I 2 c ) 48 U2+ Now if the radius of the geometrical shadow be.o, then o = (a + b) r/a, and y- - x? XIV.] DIFFRACTION OF LIGHT. 191 If g' be the distance of a point of the illuminated area upon the other line y = z/ we have evidently ~~,= ~ As special cases of these lines we have z= 0, or the axis of y, y =0 or the axis of z, and y=z. The last is dotted in the diagram, and by the result just stated corresponds to the edge of the geometrical shadow. The intensities for points along the first line are the intensities at the axis of symmetry for different radii of the orifice, or with constant radius for different values of b, the distance of the screen from the orifice. Those for points along the second line are the intensities for the case of Fraunhofer, already fully considered. In the first case we have by 20, since z = 0, U = sin ~y, U2 = 2 sin2 jy so that M2 _ (2) (U2 + U2) (sin Y) 1,~ 49 This is the expression for the intensity at a point of a screen, produced by diffraction through a narrow slit, ly in that case denoting 27ra/\f, where a is the half breadth of the slit, f the distance of the illuminated point from the slit, and 4 the distance of the point from the geometrical image of the slit on the screen. Thus Tables, which have been prepared for the calculation of the brightness in the latter case, are available also for calculating the brightness at the centre of the geometrical image of the circular orifice. The intensityis zero when -y = m7r (rn being any whole number, zero excluded), that is when the difference of path between the extreme and central rays is a whole number of wave-lengths. It is a maximum when tan ~y = jy, 7r a + b \ 7-+ b or tan - - r2 = - - r2 \an 2ab X 2ab 192 DIFFRACTION OF LIGHT. [xIv. Some values are given in the following Table. z =0. y M l=( y\ 0'000000 1'000000 4-493409 0-047190 7 '725252 0-016480 10-904120 0-008340 14-066194 0-005029 17-220753 0-003361 20-371302 0-002404 23-519446 0-001805 26-666054 0-001404 As y increases these values of ny are given approximately by the equations 2m +1 2 7r, /I 1\ r2 2m+ 1 or =+ 2 X, +a b2 2 that is the difference of path between the extreme and central rays is an odd number of half wave-lengths. The maximum intensity is then 16/y2, that is (as will be shown presently) four times the intensity at the screen due to the uninterrupted wave. For the line y = z in the diagram which corresponds to the edge of the geometrical shadow, we have by 13 1. U = sin z, U2= (J (z) - cos ), so that M2 = -1 {sin2 z + (Jo (z)- cos z)2. 50 Clearly M2 cannot vanish unless sin z and Jo (z)- cos z vanish separately, that is unless Jo (z)= 1, which is impossible unless z =0. It remains finally to find the illumination at a point on the screen when the orifice is replaced by an opaque disk, all the rest DIFFRACTION OF LIGHT. 193 of the wave being allowed to pass unimpeded. Going back to the original expressions, obtained at p. 167 above for the intensity, we see that for the total effect of the uninterrupted wave we have by I and 2 "> pdp = rr k2 12\ C, = 27r Jo (lp) cos (~kp2) pdp = -T sin S = 27r< Jo (/p) sin (i-kp2) pdp = cos Thus we get.M1 2 = 7r2 (2 = ( 2)2 52 if, as at p. 170 above, trr2 be taken as unity. This is as it ought to be, as it leads to the expression 1/(a 4 b)2 for the intensity at the point in which the axis meets the screen. We thus verify the statement, made on the last page, that the maximum illumination at the centre of the geometrical image of the orifice is four times that due to the uninterrupted wave. It might be objected that the original expressions obtained, which are here extended to the whole wave-front, had reference only to a small part of the wave-front, namely that filling the orifice. It is to be observed however that the effects of those elements of the wave-front, which lie at a distance from the axis, are very small compared with those of the elements near the axis, and so the integrals can be extended as above without error. To find the illumination with the opaque disk we have simply to subtract from the values of CO, Sx the values of Cr, Sr, given on p. 169 for the orifice. Thus denoting the differences by CG, S, we get 2 C = C - CC = - (Vosin ly - V cos Iy) 2 53 S = So - S, = - ( Vo cos ly + V, sin vy)J and M2 ) (V2 + V2), 54 or M2 = ( {+ U2 + Uo- 2 U sin l (y + ) +2U cos (y + )\ 55 G M 13 G. M. 13 194 DIFFRACTION OF LIGHT. [XIV. Comparing these with the expressions on p. 170 for M2 we see that they are the same except that now U1 is replaced by V7 and U2 by -V0. If z = 0, that is if the point considered be at the centre of the geometrical shadow, V0 = 1, V1 =0, and = (,)2 56 that is the brightness there is always the same, exactly, as if the opaque disk did not exist. This is the well-known theoretical result first pointed out by Poisson, and since verified by experiment. For any given values of y and z Ml is easily calculated from those of U1, U2 by the equations 9 o =cos I (y + ) + U2, V,=sin i(y~)- U A valuable set of numerical Tables of M2 all fully illustrated by curves will be found in Lommel's memoir. When z is continually increased in value the equations Vo=cos (y+ ', V= sin ~ (y+) more and more nearly hold, since, by I I, UC and U7 continually approach zero. The value of M2 thus becomes 4/y2 at a great distance from the shadow, as in the uninterrupted wave. As before we can find the conditions for a maximum or minimum. Differentiating, and reducing by 18 and 14, we obtain __M2 /2\2 - = - 2 (2 (z) V0. 57 The maxima and minima have place therefore when J (z) =0, or V = 0. The roots of these equations are the values of z which satisfy aJo(),O. y av O --- — = -, -- = o, az z az DIFFRACTION OF LIGHT. 195 and are, therefore, values of z which make J0 (z) and V1 a maximum or minimum. The roots of J1 (z) = 0 are given at the end of this book; those of V = 0 can be found by a formula of interpolation similar to, and obtained in the same way as, 43 above. The tangent of the inclination of the curves V0 = 0 to the axis of z is given according to 24 by 2z- V_ dy Y dz zv 2 58 By using in this the values Y \\ -V-l=- Jl- - Ji+ -.... we see that if y = oo, dz =, that is the curves are for great values of y parallel to the axis of y. Also since o=J-(o- J +..., the asymptotes of these curves are the lines Jo () = 0, drawn parallel to the axis of y. A table of the roots of this equation is given at the end of this book, and as has been seen above (p. 46) their large values are given approximately by the formula (m + ) Tr. Writing now V0=cos (Iy+ + U2 2 Y / / = cos 4 + )+ -...4 =, and making the values of y, z small, the terms after the first all disappear, and we are left with cos (y + = 0, that is y2 + z2 = (2m + 1)7ry 59 13-2 196 DIFFRACTION OF LIGHT. [XTV. This equation represents a circle passing through the origin. We infer that the branches of the curve V0 = 0 become near the origin arcs of circles all touching the axis of z at the origin. The curves V = 0 are shown in the adjoining diagram, and 24 1 4123- 567-89011 23 _ _ II7722s cI 21 2 1 2 2 - _ _+6 20 37r 18 — 17- - --- - - In-t-his]c.a 2 J or minima according as the quantity on the right is negative or positive. Hence along the line J1 (a) = 0, the intensity is a maxi= —2 - /o mum or a minimum according as J0V0 is positive or negative. On minima according as ara u h J (z) 0, or -r J (z) Vm 0 1 2 give (e +axia and minia of - te intensit crve aledydsciedfrth iarm tp.15 In this cas ^M /\2( y 1 6^ 2 ^ ^^ *(J-*46 sotatpitsi wiha iedrw prlelt heaiso ars th iarmcusth iesJ ()=0,V =0 orepodt mxm is negative or positive. DIFFRACTION OF LIGHT. 197 aM2 Where both J, (z) =0 and V = 0 the value of -I vanishes, az2 a3MM2 but not so that of a Z Hence at such points the curves of intensity have points of inflexion, but there are no others. As in the other case, points of inflexion of the intensity curve can only exist outside the shadow region of the diagram. For since J1 (z) = O, V1i = - V, 58 becomes z d= _- Y dz 1 2('which is positive if y < z, negative if y > z. But by the diagram dy i is positive everywhere. Hence there can be no intersection of the line J1 (z) = 0 with Vo = 0, except when y < z. Thus the statement just made is proved. Lastly, for the sake of comparing further the case of the disk with that of the orifice, let us contrast the intensity along a line y = cz with that along the line y = z/c. Accenting the quantities for the second line, we can easily prove that V2 + V2 + v?2 + v'1 = U2 + U2 + U'+ + U2 o(Z) cos ~, C + ). 6i ^ ~LT ^o2 + ^2-CrTT - n2 LTT+ 2~ + 2Jfo(z)cosz c+). 61 Now we have for the orifice 1 2\2T 1112 =- (U u + U2), 1/2 = c- (-) ( UT21 + UL2), and for the disk M2= 2(-) (vo + V2), \ / M2 = c2 ( Q/ + v;2). Thus by 6I c2 (M - M2i) + (12 - 12) = Jo (z) cos z c +). 62 02 Z2 198 DIFFRACTION OF LIGHT. [xiv. The shadow region is that for which y >z, and is bounded therefore by the line y = z. On this line 1M= M', M] = Ml and c = 1, so that 62 becomes 2 (M - M2)y=z - Jo (z) cos z. It is clear from the diagram that as y increases the number of dark rings which fall within the shadow also increases. The reader must refer for further information on these cases of diffraction to Lommel's paper, which contains, as we have indicated, a wealth of numerical and graphical results of great value. The discussion given above is in great part an account of this memoir, with deviations here and there from the original in the proofs of various theorems, and making use of the properties of Bessel functions established in the earlier chapters of this book. The same volume of the Abhandlungen der Konigl. Bayer. Akademie der Wissenschaften contains another most elaborate memoir by Lommel on the diffraction of a screen bounded by straight edges, in which the analysis is in many respects similar to that used in the first paper, and given above. We can only here find space for some particular applications therein made of Bessel Functions to the calculation of Fresnel's integrals, and a few other results. From the result obtained above for Fraunhofer's interference phenomena, namely that the intensity of illumination is pro4 portional to - J2 (z), the source of light being a point, we can find the intensity at any point of the screen when the source is a uniform straight line arrangement of independent point-sources. Let the circular orifice be the opening of the object-glass of the telescope which in Fraunhofer's experiments is supposed focused on the source of light. If the source is at a great distance from the telescope we may suppose with sufficient accuracy that the plane of the orifice is at right angles to the ray coming from any point of the linear source. Let rectangular axes of I, r be drawn on the screen, and let the line of sources be parallel to the axis of r and in the plane = 0. A little consideration shows that the illumination at any XIV.] DIFFRACTION OF LIGHT. 199 point of the screen must depend upon ~ and (constant factors omitted) be represented by J (z But if r be the radius of the object-glass, and ' the distance of the point considered from the axis of the telescope, 27rr Z-= b =x say, 2 and 2 = 2_ 2 2 z dz z dz Hence drd, 2 9 H 2 - v'2 if v2= p22. The integral is therefore I f Jf(z)dz K/V z _z- V2 This integral may be transformed in various ways into a form suitable for numerical calculation. The process here adopted depends on the properties of Bessel functions, and is due to Dr H. Struve. Another method of obtaining the same result will be found in Lord Rayleigh's Wave Theory of Lightt. Before, however, we can give Struve's analysis we have to prove three lemmas on which his process depends. The first is a theorem of Neumann's and is expressed by the equation 2 r2 7r J2 (z) =- J2 (2z sin a) da. 63 By 72, p. 28 above, we have Jo (2c sin - = J2 (c) + 2J1 (c) cos C + 2J^ (c) cos 2aC.... But Jof 2c sin )- cos (2csin sin q) d 2 or 2 sin ~ vr = - {J, (2c sin 6) + 2J2 (2c sin O) cos a r2(2csin)cos2...} + 2J, (2c sin ~) cos 2a +...} do * Wied. Ann. 16, (1882), p. 1008. + Encyc. Brit. 9th Ed., p. 433. 200 DIFFRACTION OF LIGHT. [xiv. by 40, p. 18 above. Identifying terms in the two equations we obtain Jn (c) = - ( J (2c sin b) do, 7 o 2 2r or J (z) =- J *d(2z sin a) da, 7r o if we write z for c and a for b. Thus the first lemma is established. The second lemma is the equation 2 J') sin (xz),, Jo (vx) = - | Vi - dz, 64 v > 0. If P, denote the zonal harmonic of the nth order, then it is a theorem of Dirichlet's that 7r = l fcos lcos n. sin -s cos n d 7 P (cos 0) = -- 2- do + 2, td 2 o c -oJ o /2 (cos -cos 0) o /2 (cos 0 -cos b) f0 sin - sinnb dG + cos sinn d = -,-I - ' - do +,f, ' do. Jo2 (cos& -cos o) Jo V2 (coos )Subtracting the second expression on the right from the first we obtain 0 cos (n- ) r sin(n-2)h d 2) do -n do. Jo2 (cos - cos 0) J o 2 (cos 0 - cos h)) For 0 and b put 0/n, 0/n, and let n be made very great; the last equation becomes 0 cos fdf _ r sin f dc JoV 92-02 J _ 02 The quantity on the left is 7rJo (0), (see p. 32 above). Hence j: sin Od Write for, and for, for d and 65 becomes Write xz for G, and vx for O, xdz for dob and 65 becomes r o() =f sin (x) dz 2 whi2 _ which is the second lemma stated above. XIV.] DIFFRACTION OF LIGHT. 201 The third lemma is expressed by the equation [i T / * \ * ^ sin z Jo(z sina) sin ada== 66 Using the general definition (p. 12 above) of a Bessel function of integral order we get a Jo (z sin a) sin ada = (_s- () sin2+1ada (-)s /(z82 22 (11s)2 (ls)22 n l(2s + 1) = (-)Sz2S sin z II (2s +1) z which was to be proved. Returning now to the integral / J (z) dz v Z V/2 V2 let us denote it by Z. We have by the first lemma z= - = --- f J (2z sin a) da. 7qt v Z VZ2 _ V2 0 But by 20, p. 13 z siln a J2 (2z sin a) = --- {J1 (2z sin a) + J3 (2z sin a)}, and by 45, p. 18 J1 (2z sin a) = - sin (2z sin a sin /) sin / d/3, rJ o J3 (2z sin a) = - sin (2z sin a sin /) sin 33d/,3, 7r o so that = sin a da (sing + sin 3/3) d/3 sin 2sin a si ) dz r-/, Jo ov ~ -vy2 But if we put 2 sin a sin / = x we get by the second lemma 0 sin (2z sin a sin /) dz r' J, - - 2 = Jo (vx). z2 - 2v2 2 Hence Z= 2- (sin /3 + sin 3/) d/ J0 (2v sin a sin /) sin ada, 202 DIFFRACTION OF LIGHT. [XIV. which by the third lemma becomes 1 2 ^ sin (2v sin }) Z= (n + ) sin + sn 3) s) dfl rrJ o 2v sin / 2 [rf = - sin (2v sin /3) cos2 / d/. 67 T'V Jo Let now H0 (z) be a function defined by the equation 2 fgr Ho (z) = - sin (z sin 0) dO. 68 7TJ 0 Expanding sin (z sin 0) and integrating we obtain 2( z3 z5 1.- z132+ 123 5.-..}. 69 Now let H1 (z) be another function defined by H, (2) = Ho (z) zdz, then by the series in 69 d (z) =12-..d-i-p.. '2 7 } 70 We shall now prove that H1 (z) = 2Z- sin (z sin 0) coss 0 dO. 71 7T JO It can be verified by differentiating that d ( d\ 2 z dz z o H (z) = - - Ho (z). 72 z dz \ /dz) TZ7r'Z Multiplying by zdz and integrating we find 2z dEHo (z).... z. 73 7r dz Now by 68 dHo(z) 2z 'IT zd ()- = - cos (z sin 0) sin 0dO. dz 7T Jo Hence H (z)-= 2z - cos (z sin 0) sin d0} =-f {l-cos (z sin 0)} sin o0d = sn 0 n z0 74 =7 sin (z sin 0) sin 0 dO. 74 7rdJO DIFFRACTION OF LIGHT. 203 It may be noted that every element of this integral is positive. It is clear from the form of Hi (z) given in the first of the three equations just written that H1 (z) approximates when z is large to 2z/r. Integrating 74 by parts we obtain H1 (z) = - sin (z sin 0) cos2 dO, since the integrated term vanishes at both limits. If we write 2v for z and 3 for 0 the equation becomes H (2v) = 2 (2v)' fsin (2v sin 3) cos2 i3df3. 75 7r o Hence H (2v) 2 f7 U4v - v sin (2v sin /) cos2 /d/ = Z, by 67. It is to be observed that the functions here denoted by Ho (z), H1- (z) are the same as Lord Rayleigh's K (z), K (z) discussed in the Theory of Sound, ~ 302, to which the reader is referred for further details. The function H1 (z) differs however from the function H, (z) used by Struve. If we denote the latter by i1 (z), the relation between the two functions is H1 (z) = Z1z (). The value of H, (z) can be calculated when z is not too great by the series in 70, but when z is large this series is not convenient. We must then have recourse to a semiconvergent series, similar to that established in Chap. Iv.-above for the Bessel functions. The series can be found easily by the method of Lipschitz, already used in Chap. vii. The following is a brief outline of the process *. By the definitions of the functions we have o) ~o( )2 7r. J,(z) - iH(z)=- I e-izs"'OdO 7rJo 2 rl e-iv, =- I --- dv.' See Theory of Sound, ~ 302 See Theory of Sound, ~ 302. 204 DIFFRACTION OF LIGHT. [XIv. Now take the integral r e-w dw J/1 + w2 (in which w = u +iv) round the rectangle, the angular points of which are 0, h, h + i, i where h is real and positive. This integral is zero, and if h = cO it gives after some reduction I p-izv 32 — \ -.dv= l ^l+ _ d_ 8 -v/1- +2 Z o z2 e- (z-4(' ) e diS Expanding the binomials and integrating, making use of the theorem f e-3-d3 = n11 (q -), and equating the real part of the result to 7-TrJo (z) and the imaginary part to - i7rHo (z), we get the expansions required, namely Jo(2) as in Chap. iv. and Ho(z) = 2 (z-1 z- + 12. 32Z-5 _ 12. 32 5-7 +. ) /2 + A/- {P sin (z - r) - Q cos ( - 7) 76 +Z 4 4: 76 where P and Q have the values stated on p. 48 above (z being written for x). From this the value of H, (z) is at once found by the relation 73 and is H, ()= _ (z + 2-1 _ 3Z-3 + 2. 32.,-5 -...) 7T' /2 os Z ) - (12 - 4) (32 4) -yV,cos (z- )^l — 47 (12 - 4) (32 - 4) (5 - 4) (72 -4)_ 1.2. 3.4.(8z)4 4" 4- V' sin (z - ) {1-4 (1p _ 4) (32 - 4) (5_ 4) 77 1.2.3.(8z)3 **- i XIV.] DIFFRACTION OF LIGHT. 205 It is to be noticed that H,~(2v) is nowhere zero, and that H, (2v)/v3 has maxima and minima values at points satisfying the equation d H, (2v) 4v2 Ho (2v) - 3H v (2v) 0 dv v3 v4 7 The corresponding values of v are therefore the roots of 4v2 (2v)-3H1 (2v) = 0. Now let there be two parallel and equally luminous linesources, whose images in the focal plane are at a distance apart v/l= Tr/p, say. It is of great importance to compare the intensity at the image of either line with the intensity halfway between them. In this way can be determined the minimum distance apart at which the luminous lines may be placed and still be separated by the telescope. We shall take the image of one as corresponding to v = 0, and that of the other as corresponding to v = 7r. Thus the intensity at any distance corresponding to v is proportional to 4(v) Putting 7T H,(2v) L = 2 (2v) we have by 70 1 22v2 24v4 L(v)12.3 12. 32 5 +12. 32. 52.7 The ratio of the intensity of illumination midway between the two lines to that at either is therefore 2L (l7) L (0) + L (7r)' This has been calculated by Lord Rayleigh (to whom this comparison is due) with the following results L(0)= -3333, L(7r) = 0164, L (7r)= 1671, so that 2L (7r) zwN\,, - '955. 79 L (0) + L(7r) The intensity is therefore, for the distance stated, only about 4 — per cent. less than at the image of either line. 206 DIFFRACTION OF LIGHT. [XIV. Now 27rr Xb which gives t_ - b 2r Since b is the focal length of the object-glass, the two lines are, by this result, at an angular distance apart equal to that subtended by the wave-length of light at a distance equal to the diameter of the object-glass. Two lines unless at a greater angular distance could therefore hardly be separated. This result shows that the resolving (or as it is sometimes called the space-penetrating) power of a telescope is directly proportional to the diameter of the object-glass. By multiplying 2 H, (2v) wr (2v)3 by Judd, that is by dv, and integrating from =-oc to = + oo we get an expression which, to a constant factor, represents the whole illumination received by the screen from a single luminous point the image of which is at the centre of the focal plane. Or, by the mode in which H1 (2v)/v3 was obtained, it plainly may be regarded as the illumination received by the latter point from an infinite uniformly illuminated area in front of the object-glass. If the integral is taken from: to + cc it will represent on the same scale, the illumination received by the same point from an area bounded by the straight line parallel to iJ corresponding to the constant value of ~. The point will be at a distance 4 from the edge of the geometrical shadow, and will be inside or outside the shadow according as f is positive or negative. We have by 71 f H,(2v) d'1 = o d sin (2v sin /) dv = - / cos 13d — dv Jo (2v)3 J o - fcoS2/3d/3= - Now Hi(2v) A- HI (2v) -dv - (2v) dv (2v. (2v)3 0(2v) DIFFRACTION OF LIGHT. 207 The second term on the right can be calculated by means of the ascending series 70. Hence we get H(2v) tr 2 ( v 22v J, (2v)3 8 r 12.3 12.32.3.5 24v ) + 2. 32.2. 5.77 } 80 This multiplied by 4/?r is the expression given by Struve for the intensity produced by a uniform plane source, the image of which extends from v to + oc. For the sake of agreement with Struve's result we write when v is positive r/ V1 +1 2 4 2v (2v) ( ) = I 3 3 _ + } 8I 2 -2 12 1[3 1".32. 3. 5 Hence if I be the illumination when the plane source extends from - oo to + oc we have I(+ v)+ I(-v) =I=1. This states that the intensities at two points equally distant from the edge of the geometrical shadow, but on opposite sides of it, are together equal to the full intensity. The intensity at the edge of the shadow is therefore half the full intensity. The reader may verify that when v is great the semi-convergent series gives approximately 2 /l 1 d 1 cos(2v+47r) I(v)=-2 + 123 2 vcos2 The following Table (abridged from Struve's paper) gives the intensity within the geometrical shadow at a distance: = bXv/27rr from the edge, and therefore enables the enlargement of the image produced by the diffraction of the object-glass to be estimated. 2xrr V bX I (-v) = 1 - I(+ v). v I(+v) v I(+v) v I(+v) 0'0 -5000 2 '1073 7 -0293 0'5 a 3678 3 -0630 9 '0222 1 0 -2521 4 -0528 11 -0186 1 5 -1642 5 -0410 15 -0135 208 DIFFRACTION OF LIGHT. [xIv. We shall now consider very briefly the theory of diffraction of light passing through a narrow slit bounded by parallel edges. We shall suppose that the diffraction may be taken as the same in every plane at right angles to the slit, so that the problem is one in only two dimensions. Let a then be the radius of a circular wave that has just reached the gap, and consider an element of the wave-front in the gap. Let also b be the distance of P from the pole so that its distance from the source is a + b, ds the length of the element of the wave, and 8 the retardation of the secondary wave (that is the difference between the distances of P from the element and from the pole). The disturbance at P produced will be proportional to os27 (t ) ds. If the distance of the element from the pole be s, and s be small in comparison with b, then it is very easy to show that a + b2 a+b Writing as usual ~^vr2 for 27r8/X, we get 27r 8 7rr (a + b) - ~7 == -v-b-s x 2 abX The disturbance at P is therefore /tV I72\1 t t cos 27r ( - = cos V2 c 27 + sin rv2 sin 27r. The intensity of illumination due to the element is therefore constant, being proportional to cos2 7rV2 + sin2 r2,rv where 2 2 (C + b) 2 abX The whole intensity is thus proportional to { (cos rv2. dv} + {/sin 17T2. d} the integrals being taken over the whole arc of the wave at the slit. DIFFRACTION OF LIGHT. 209 The problem is thus reduced to quadratures, and it remains to evaluate the integrals. We shall write C= Ccos r2TV2dv, S== sin lrv2dv. Jo /o C and S are known as Fresnel's integrals. Various methods of calculating these integrals have been devised; but the simplest of all for purposes of numerical calculation is by means of Bessel functions, when Tables are available. Let -7rV2 = z, then 2 Let us now consider the Bessel function integrals on the right. Using the relation J' (z)2 (-1 - J (z)-Jn+l (z)) we have J-_ (z)= 2JI (z) + J) (Z) = 2i (z) + 2J (z) +... + 2J;+1 (z) + J4n+s (Z) 2 = 2 22 Thus we obtain - J $__1 (,) d, = J2 (,) + Ja (z) +.. +&J;1(Z) + S |Jn+3- ( *) d,. 84 Jo0 2 df2J 3 By taking (4n + 3)/2 sufficiently great the integral on the right of 84 may be made as small as we please. Thus we get C = fi S (X) dz= J () J+ J ( Z)+ J (z) + 85 Similarly we find S= fJ (z = J3 () () + $ () + J()+ 86 Jo 2 These series are convergent, and give the numerical value of the integrals to any degree of accuracy from Tables of Bessel functions of order (2n + 1)/2, by simple addition of the values of the G.M. 14 210 DIFFRACTION OF LIGHT. [XIV. successive alternate functions for the given argument. The series are apparently due to Lommel, and are stated in the second memoir referred to above, p. 198. He gives also the series 7=- f J_, (z)dz = -/2 (P cos 2z + Q sin ~z) 87 f= Jt (z) dz = \/2 (P sin 1z -Q cos 1) 88 0 where P =J1 ( )- J (z) + J- (Z)-., Q =3 (z)- J )++ J (z) - The proof is left to the reader. C and S were expressed long ago in series of ascending powers of v by Knochenhauer, and in terms of definite integrals by Gilbert. From the latter semi-convergent series suitable for use when v is large are obtainable by a process similar to that sketched at p. 203 above. It is not necessary however to pursue the matter here. The very elegant construction shown in the diagram, which is known as Cornu's spiral, shows graphically how the Y value of C S2 varies. The abscissae of the curve are values of C and the ordinates values of S. It can be shown that the distance along the curve from the origin to any point is the value of v for that point, that (B the inclination of the tangent to the axis of abscissae is 27rv2, and that the curvature there is 7rv. As v varies from 0 to oo and from 0 to - oo the curve is wrapped more and more closely round the poles A and B. The origin of the curve corresponds to the pole of the point considered, so that if vi, v2 correspond to the distances from the pole to the edges of the slit, we have only to mark the two points v1, v2 on the spiral and draw the chord. The square of the length DIFFRACTION OF LIGHT. 211 of this chord will represent the intensity of illumination at the point. The square of the length of the chord from the origin to any point v is the value of C2 + S2, that is of f rz }2 rz 2 1 i J, (z) dz +I {f J z) dz. 0 0 As v varies it will be seen that the value of this sum oscillates more and more rapidly while approaching more and more nearly to the value -. 14 —2 CHAPTER XV. MISCELLANEOUS APPLICATIONS. IN this concluding chapter we propose to give a short account of some special applications of the Bessel functions which, although not so difficult as those already considered, appear too important or too interesting to be passed over entirely or simply placed in the collection of examples. We will begin with the equation at which occurs in various physical problems, such as the small vibrations of a gas, or the variable flow of heat in a solid sphere. Using polar coordinates, u is a function of t, r, 0, b, such that Gu 2 2a2tt at 1 a (. as \ 1 a2+S = r- ir 2 + 2r +s i (sin 0 +s i a -2r ar ar sin 0 a1 a2 Assume, as a particular solution, t = e- K2a2tvq, where v is a function of r only, and Sn is a surface spherical harmonic of order n, so that 1 r / ___ 1 a2Sn sin 0 aD ysin 0 I+ + si2 0 ~ (iZt + 1) S. = 0. Then after substitution in (2), it appears that v must satisfy the equation d2v 2 dv {2 n (n + 1l) =; dr2 r + dr r and now if we put v =r ~-o, MISCELLANEOUS APPLICATIONS. 213 we find that w satisfies the equation dw l dwu (n + i) 7ow + - _ + -— ) 0. dr2 r dr r2 Hence = AJ,, (Kr) + BJ__ (cr), and finally -Ka e2t {AJ+ (Kr) + BJ l (K )} 3 is a particular solution of the equation I or 2. In practice n is a whole number, and by a proper determination of the constants n, K, A, B the function U= E e- AJ1+ (cr) + BJ_1 ( Kr) 4 is adapted in the usual way to suit the particular conditions of the problem. (See Riemann's Partielle Differentialgleichungen, pp. 176-189 and Rayleigh's Theory of Sound, chap. xvIi.) If in the above we suppose n = 0, the function S,o reduces to a constant, and 2 J_ (Q)= sin Ku (see p. 42); thus with a simplified notation, we have a solution sin Kr _ -t U=:TA slnsec -K2a2t 5 K r which may be adapted to the following problem (Math. Tripos, 1886): "A uniform homogeneous sphere of radius b is at uniform temperature Vo, and is surrounded by a spherical shell of the same substance of thickness b at temperature zero. The whole is left to cool in a medium at temperature zero. Prove that, after time t, the temperature at a point distant r from the centre is 4 sin icb - cb cos Kb sin cr -K2n2t x 4Kb- sin 4cb r 214 MISCELLANEOUS APPLICATIONS. [XV. where the values of K are given by the equation 2Kcb tan 2Kb.=_. 1 - 2hb' h being the ratio of the surface conductivity to the internal conductivity." Here the conditions to be satisfied are V= V0 from r= 0 to r= b, V=0... r=b...r=2b, when t = 0; and - +hV= O when r = 2b, for all values of t. Now, assuming a solution of the form V = IA sin Kr eK20 r the last condition is satisfied if a Ksin r\ h sin 1cr or\ r + r when r = 2b: that is, if Kc cos 2Kcb sin 2cb h sin 2cb 6 - ~W -+ - 0, 2b 4b2 2b leading to 2ccb tan 2Kb 2=1 - 2hb' as above stated. Proceeding as in Chap. VI. above, we infer that A,, sin2 crdr = Vr sin Krdr, that is, AK b - J V r sinrdr = (sinb b cos cKb and hence 4 sin fcb - Kcb cos Kb K ' 4cb - sin 4Kcb which agrees with the result above given. XV.] MISCELLANEOUS APPLICATIONS. 215 Returning to the solution given by equation 4 above, we may observe that when a2 is real and positive, the solution is applicable to cases when there is a "damping" of the phenomenon considered, as in the problem just discussed. When there is a forced vibration imposed on the system, as when a spherical bell vibrates in air, we must take a2 to be a pure imaginary + ia/K, so as to obtain a time-periodic solution. The period is then 27r//ca. An illustration of this will be found at the end of the book. We will now proceed to consider two problems suggested by the theory of elasticity. The first is that of the stability of an isotropic circular cylinder of small cross-section held in a vertical position with its lower end clamped and upper end free. It is a matter of common observation that a comparatively short piece of steel wire, such as a knitting-needle, is stable when placed vertically with its lower end clamped in a vice; whereas it would be impossible to keep vertical in the same way a very long piece of the same wire. To find the greatest length consistent with stability, we consider the possibility of a position of equilibrium which only deviates slightly from the vertical. Let w be the weight of the wire per unit of length, /8 its flexural rigidity. Then if x is the height of any point on the wire above the clamped end, and y its horizontal displacement from the vertical through that end, we obtain by taking moments for the part of the wire above (x, y) d/ = J w (y' -y) dx', I being the whole length of the wire. Differentiate with respect to x; then ' dx 3 w( dx) dx' or - dy(-x) =, if =p dx 216 MISCELLANEOUS APPLICATIONS. [XV. Now put -x = r3. then dp 3 dp dx 2 dr' d2p 9 f 2dp 1 idp' dCe2 4r d2 +r 3 d and the transformed equation is d2p 1 dp 4w dr 3r dr + 9P0; and now, if we put p =- z, it will be found that d2z 1dz 4w1I dr2 + dr + 9 -9r z =O Hence if 24w sc 9/3' it follows that z = AJ, (cr) + BJ_ L (cr). When x= 1, that is, when r = 0, we must have dp dx- whence rdp dr when r= 0; that is, i dz1 _2 ), r3 r3 3+r =~, or dz 3r- + z - = 0. 3r3 Now the initial terms of JI (cr) and J_ (Kr) are of the forms XV.] MISCELLANEOUS APPLICATIONS. 217 J, (,r)= ar+ + +..., J_ (cr) = ' r - ~ +/ 'rs +..., and it is only the second of these that satisfies r- 33r-+z =0, V dr / ' when r = 0. Therefore A = 0. Again, when x = 0, that is, when r = lt, p, and therefore z, must be zero. Hence, in order that the assumed form of equilibrium may be possible, J_ ) (Kcl) = 0. The least value of I obtained from this equation gives the critical length of the wire when it first shows signs of instability in the vertical position; and if I is less than this, the vertical position will be stable. It is found that the least root of J_ (x)= 0, is approximately 1'88: so that the critical length is about (188)t or 1 996 4/j3w, approximately. To the degree of approximation adopted we may put z = 2 ///w, or in terms of / and W, the whole weight of the wire, I = J18/ w = 283 s//3W. Of the two formulae given the first is the proper one for determining the critical length for a given kind of wire; the second is convenient if we wish to know whether a given piece of wire will be stable if placed in a vertical position with its lower end clamped. (See Greenhill, Proc. Camb. Phil. Soc. IV. 1881, and Love, Math. Theory of Elasticity, II. p. 297.) 218 MISCELLANEOUS APPLICATIONS. [XV. As another simple illustration derived from the theory of elasticity, we will give, after Pochhammer and Love (I.c. p. 115), a short discussion of the torsional vibration of an isotropic solid circular cylinder of radius c. If (r, 0, z) are the coordinates of any point of the cylinder, and u, v, w the corresponding displacements, the equations of motion for small vibrations are 8a2I aA 21/ 8wa3 8_2 Pa= (X + 2 -) - + 2~ P 'ar ~z r ar dz a2= + 2) Iaa 2 a8 8 + 2. a3 = (at2 2~u) r a az 2 a2W aA 2/, a 2, aw, — at2 = (\ 4 2-) a (rat) + 2/ a+ az r a r o where 1 a(ru) 1 av aw A...+- + — r ar ra0 a' _1 8w 8 (rv) l 2'1= r Vo a azl atn aw 2 as ar - 3r ( r a) The stresses across a cylindrical surface r = constant are P,, = X + 2/, Por = I a- r ra, r +' Pzr = / c + ar We proceed to construct a particular solution of the type u =, w= O, v= VeTi(Yz+t), where V is a function of r only, and y, p are constants. In order to obtain a periodic vibration with no damping, we suppose that p is real. The torsional character of the oscillation is clear from the form of u, v, w. MISCELLANEOUS APPLICATIONS. 219 If we put, for the moment, ei (y+pt) Z we have A'-0, 271 = - iyVZ, 2W'2 = 0, 2 - Z a (r V) r ar and the equations of motion reduce to two identities and - pp2VZ = - 2VZ + Z dr( (rV)) Thus V must satisfy the equation d2V I dV ( -/y'l)2 2 de d- V+ _= 0t dr2 r dr +( r2) and since V must be finite when r = 0, the proper solution is V= AJ1 (r) where C2 = P p2 2_ 2 If the curved surface of the cylinder is free, then the stresses Prr, Per, Pz, must vanish when r = c. Now Pr, and Pz, vanish identically; Por will vanish if (I {Ji (Kr)} 0, dr r when r=c: that is, if KcJ1 (/c) - J (KCC) = 0, or, which is the same thing, if IccJ2 (Kc) = 0, (see pp. 13, 179). If /c = 0, the differential equation to find V is d2V 1 dV 1 dr2 + r dr r of which the solution is V=Ar+ B r 220 MISCELLANEOUS APPLICATIONS. [XV. or, for our present purpose V=Ar. This leads to a solution of the original problem in the shape u = O, wu = 0, v = Arei(yz+2t), with PP' - 72 = 0; and in particular we have as a special case V =,1. ^^COS 777^ n7rt 1k n7vt /4 v sin 1 cAos — +B sin - when n = 0, 1, 2,..., and 1 is a constant. But we may also take for Kc any one of the real roots of J2 (cc) = 0 of which there is an infinite number. If Ics is any one of these, p any real constant, and 7 determined, as a real or pure imaginary constant, by the equation 7=P 2 ry2=-p_ we have a solution v = AJ, (K)r) ei(Yz+t). Unless some further conditions are assigned, p is arbitrary, whatever value of Kc is taken: that is to say, vibrations of any period are possible. When the period is 2rr/p the velocity of propagation parallel to the axis of the cylinder is P A/Atk 2 _ Jpp 2 which is approximately equal to ^//t/p so long as _tK2 is small compared with pp2. If pp2- _ g2 is negative the type of vibration is altered: there is now a damping of the vibration as we go in one direction along the axis of the cylinder. Special solutions may be constructed to suit special boundary conditions: thus for instance if we put xv.] MISCELLANEOUS APPLICATIONS. 221 v = - (Asm, cos pt + Bsn, sin pt) J1 (Ksr) sin I s, 3n2 when m is a real integer, Kc, any root of J2 (Ks,) = 0, and p2 = 1 2 + it 7 } this gives a possible mode of vibration for a cylinder of radius c and length 21, the circular ends of which are glued to fixed parallel planes, the curved surface of the cylinder being left free. The doubly infinite number of constants Asm, B,,, have to be determined by suitable initial conditions. For the discussion of the extensional and flexural vibrations the reader should consult Love's treatise already referred to, and the memoir of Pochhammer, Crelle, lxxxi. p. 324. Many other illustrations of the use of Bessel functions in the theory of elasticity will be found in recent memoirs by Chree, Lamb, Love, Rayleigh and others. To conclude, as we have begun, with the oscillations of a chain, let us modify Bernoulli's problem by supposing that the density at any point of the chain varies as the nth power of its distance from the lower end. Proceeding as on p. 1, but measuring x from the free end, the equation of motion is " dy _ d (gzx+m dy dt" dx \n + dx' and if we put y = u cos 27rpt, where u. is a function of x, we have gx d2jt dau n+l -T-. + 9 - + S 47rpt = 0, n + 1 dx - dx + or d2 it+ + 1 dzi C2 + - - - -u = 0, d x dx x where IC = 27rp /(n + 1)/g. Assuming U = (t0 + C1a + a2x2 +.... 222 MISCELLANEOUS APPLICATIONS. [XV. we find that the differential equation is satisfied by }dX _2 _ C4X2 =a {1 n+l - + 2()(n+ 1)( +2) fiC6X3 2.3 (n+ 1)(n + 2)(n +3) + ' or, which is the same thing, by = Ax-~nJ, (2Kc2) Finally, therefore, y = Ax-J {47rpV(n } cos 2) pt. =A `V pcos 27rpt. Professor Greenhill, to whom this extension of Bernoulli's problem is due, remarks that to realise the conditions of the problem practically, we should take, instead of the chain, a blind composed of a very large number of small uniform horizontal rods, the shape of the blind being defined by the curves n-iy = + xn, with x positive. Thus n = 1 gives a triangular blind, and so on. In connection with the reduction of the differential equation at p. 216, it may be pointed out here, that, if yr-" be substituted 1 for u, and x1c-l1 for r, the differential equation of the form d21 1 du 1 dr2 + (2X + 1) r d + (/C22 + -_ ) = 0, can be reduced to the standard form d2y l dy n2) dx- x I -x x so that it is integrable by Bessel functions. This gives a general rule for the transformation in cases in which it is not so obvious as in the case considered above. In that case u=p, x=-~, n = +, = 1, KC2 = 4W/9,8. (See p. 233 below for other examples.) NOTE. As the determination of the coefficients of IY(x), J,,(x) in that second solution of the general differential equation which vanishes at infinity is not without difficulty, and the solution is of great importance for physical applications, the following explanation of Weber's treatment of the problem (Crelle, Bd. 75, 1873) may not be superfluous. It has been proved, pp. 59, 60 above, that if z be complex, and such that the real part of zi is negative +1, f eZ^ (1 - X2)%-i d = 22n irrI (n - ) JJ (z). I J-1 Further the investigation given in Chapter viI. above shows that the differential equation dz2 dz \ 2 / is satisfied also by taking w- f ez (X2 - 1) -1 dX. 2 For by differentiation it will be found that the quantity on the left in the differential equation reduces to -iz'- {eiZX (X2- 1) which vanishes when = 1, and when = oo. These two definite integral solutions of the differential equation seem to be due to Riemann, who gave them in his memoir on Nobili's Rings referred to on p. 128 above. In the second of these solutions, on the supposition that the path of integration is along the axis of real quantity from 1 to oo, the real part of iz must be negative, so that, either z must be essentially complex, or n < 0. The solution then in the absence of fulfilment of the latter condition does not hold for real values of z. We can find a solution which holds for both real and complex values of z as follows. 224 NOTE. Let a new path of integration from + 1 to 0, and from 0 to oo i be chosen. We have =n { eiz (X2 1) dX + eZ (X2 -)n-d. 3 Now, for the moment let z be real, = x, say, and we have putting in the second integral /Li for A, where /J is real w= (- x) {1 eiZx(l -2)n dx + je- (1 +)- d. 4 W1 JAi e x4 Since X is real in the first integral we may use X as variable in the second part also, and we have Real part of w - (- x) { e~ (1 + X2)"-2lX - sin (^) (1 - X)n- dl. Thus if u = X {fsin (xA)(1-X2) d f e-(1 + ~2)2dX }, 5 u is a solution of the general differential equation. It will hold also for complex values of x provided that the real part of x is positive. We now write -= An,,Yn X) +,BnJn (), 1. 3. 5... (2n - 1) where An, Bi are constants to be determined. This can be done as follows. Multiplying both sides by xn, and putting x = 0, we have zero for the second term on the right, and therefore anU Lt -= Lt A, YA(x). =o 1. 3... (2n -- 1) =o The first term on the right of 5 multiplied by xn vanishes when x = 0. Expanding the quantity under the sign of integration in the second term by Taylor's theorem we obtain the expression n eXA X2n- +... + X2... + X2f-1_ d}, where (2n - 1)(2n-3)...3 + X2 ( 1 2n- (n-1l)! V 2/ k2(n-,) (0 < 0 < 1). Integrated this takes the form, Ir(2n) 2n - r(2nS-l) and the limit when x of is therefore - (), that is - ( - )!. and the limit when x = 0 of Xnqt is therefore- r (2n), that is - (2n - 1)!. NOTE. 225 Now from the explicit form of Y (x), 30, p. 14 above, we see that the limit of x" Yn (x) when x = 0 is - 2-1 (n - 1)!. Hence finally xT aU (2n- 1)! Lt _ _ _ = _ ____ )___ An 2n-1 (n-i1)! to 1.3... (2n-l):-1.3...(2n-l)- A'- ("-1)' or A = 1. 6 Writing u, for what we have called n, we have fi r 2n~-l~ ~ /00,2n-1 d n = J C cos (X$). (i _ )2n-2 de + | e-X (1 + 2) 2 d-1 Integrating by parts we get 2n+2 = qv~n 1 1 Z t sin (x~). (1 _ )-2- de dx x7 2? 1 2n +1 2n+ I o x 0 C^ ^ 2n+l + 2n+1 Jo e ( 2) 2d - +-2-1 =-XZCn+1 I d un, or n+l xa+l x dx x Thus un+1 is derived from us without any change of coefficient, and so comparing with the case of n =0 (see Examples 15...19, p. 229 below) we get B = y-log2. 7 Thus for the values of x specified the solution of the general differential equation may be written by 6 and 7, u:= C {Y, (x) + (y- log 2) J ()}, 8 where C is a constant. G. M. 15 EXAMPLES. 1. If, as on p. 4, we put L = - e sin 4, prove that a2 a 1 1-e2 a20 -+- - + _ = O ae2 e ae e2 8a2 and hence obtain Bessel's expression for 4 in terms of /. 2. Prove that, in the problem of elliptic motion, the radius vector SP is given by the equation a 1 r -e cos 4 = 1 + 2 {J (e) cos + 2(2eJs) cos 2t +...}, (see p. 19). 3. Verify the following expansions:(i) eO = J0 (x) + S {(n + /n)2+ 1 )s + (n - 1/n2)+ 1)s} J, (x). 1 (ii) cosh nx J0 (x) + 2S cosh sJ8 (x), [s = 2, 4,...] sinh nx= 2: sinh sf Js (x), [s = 1, 3,...] where 4 = sinh-l n. (iii) cosnx= Jo (x) + 2 (-)" cosh s Js (x), [s = 2, 4,...] sin nx= 2 (_)i(8- ) cosh s Js, (x), [s = 1, 3,...] where 4 = cosh-ln, n being supposed greater than 1. 8 1 (iv) Y3 = J3 log xZ - 3 x _ - I J3+ 3 (4s + 1) 4s + 3 2 (2s + 1) (2s + 2) s4+ 4s (2s + 3) EXAMPLES. 227 4. Prove that 1 = J+ 2J + 2J +..., J = 2 (JoJ2 + J1 3 + J2 +...), 2JJ3 - J2 = 2 (JoJ4 + JJs + J2J6 +...). 5. Show that bcJ, (b2+ c2) = 2b2 + c2 {J (b) J (c) - 3J3 (b) J () + 5J (b) () -...}. 6. Prove that _ 2n+3 _ 2n+5 3 e-(I 2.7 I! + 2 (2n + 2) 2. 3 (2n + 2) (2n +5) (2n + 7) t 2. 3. 4. (2n + 2) (2 + 4) (2n +7)(2n+9) 2. 3. 4. 5. (2n + 2) (2n + 4) 7. Verify the following results (taken from Basset's Hydrodynamics, vol. In.) Ko (ax) cos bxdx = 7r (a2 + b2) -, ~oo ~ ~ ~ ~ 1 fo -ax Ke (bx) dx (b2 a- 2)^ tan - a; b > a, a2 -; 2a, = (a2 _ b2)- tanh --; b < a, a Ko (ax) Jo (bx) dx - (a2 + b) - F {b (a2 +, <,oo J e- {Jo (bx)}2 dx = 27r- (a2 + 462)- F 1{2b (a2 + 4b2)-I}. o [F (k) denotes the first complete elliptic integral to modulus k.] 8. Prove that 2 ro Y(x) = - J cos (x cosh 0) dO, -a22 Y (bx) dx2 e-a K e-a2 Ko (bx) dx e8 K (Basset.) 15-2 228 EXAMPLES. [The following Examples (9-19) are taken from Weber's paper in Crelle, Bd. 75, 1873.] 9. Prove that J1 (x)= -I i cos cos 0de. 7 JO 10. Prove that Ji (x) Jo (ax) dx= 1, (a2 > 1) = 1, (= 1) =0, (a2> 1). [Substitute the value of J1 (x) from Ex. 9, and integrate first with respect to x.] 11. Prove that f -^AX(2 ) = f 6 e- sin (Xx) dA sin (Xr) J (r) dr. Hence show by integrating first with respect to X on the right and having regard to 151, p. 73, that Jo 2 sin (x) - and therefore fl sin (Xx) dXf sin (xc) dA Jo r = j1 — \ 1 JX21 12. Prove that 00 f o log J (x) dx =-( + log 2), where Y - = e- log xdx, (see p. 40 above). [By the last example, f ex logZJ0 (x) d r- J Ix= - | eelog x sin (Xx) dx, J/X2 1' and the second integral on the right can be evaluated by known theorems. Or, more simply, the value of log x given by the equation 00 e-~u_ -ux Jo X log x = f du may be substituted in the given integral, and the integration performed first with respect to x then with respect to u.] EXAMPLES. 229 13. Prove that co f Yo () dx = log 2 -. [Use Neumann's series for Y,, p. 22 above, and the theorem, 149, p. 73, with the result of the preceding example.] 14. Establish the equations (a) / oA k 2 +X2 k h IX2- _ ( X) sin (aX) Jo (Xx) dX sinh ka 1 _ _ k _ (b) Jo k2+X2 k X2 -( 1 J 00(b) cos (aX) J, (Xx) = e-Ch ka (c) 2 ) X2 A - - Jo (ik) A X sin aX) J' {r(x < a). ks + X2 2 () fA sin (aX) J, (x) cA ek JO (k [Use the theorems J cos XX dA7r -kz Xsin(x) kxl ko +X dx=-e, 2 + 2. 15. Prove that the definite integrals r+l eizx f eiZA. I - = d f) dA J_1 JI X2 h J A2 (in which z may be real or imaginary, but is always such that the real part, if any, of iz is negative) both satisfy the differential equation of the Bessel function of zero order. 16. If (see Ex. 15) 2 f0 e 1iz /(3)==- J. dA, prove that for z real and positive (= x, say) 2 f"cos (AX) f(x) = dA - (x) - A + iJo (x), %/("2- 1 and for z real and negative (=- x), 2 f cos (xA) _ (x) f (- Xx) = 0 j- _ - iJ0(X)4 7rJ 1 /X2-1 230 EXAMPLES. 17. If we write, as from Ex. 15 we are entitled to do, f () = A Y, (z) + BJo (z), prove that f (- x) -f (x) = A-7iJ0 (x), 2 and therefore A -- 7r by the theorem of Ex. 16. [Change from + x to- x along a semicircle round the origin and have regard to the term J0 (x) log x in Y0 (x).] 18. Prove that of (x) dx = i, and hence by Ex. 13 and the theorem I49, p. 73, that B =i — (y-log 2), 7i so that f _ - 1j dX = - { Yo (z) + (y - log 2) J (z)} + Jo (). 19. Prove that X2 (X) d= - { Yo (ikz) + (y - log 2) J0 (i/c) + 2 Jo (ikz). 20. Prove that if u is a function of x and y which satisfies the equation a2u a2u 2 + K2 =0, ax2 + ay2 = au aui and which, as well as its derivatives, y, is finite and continuous for all points within and upon the circle x2 + y2 _ r2 = 0, then udO = 2TtruJo (Kr), when the integral is taken along the circumference of the circle, and uo is the value of u at the origin. (Weber, Math. Ann. i. p. 9.) EXAMPLES. 231 21. Prove that if V 27r' f d/ ed-z cos Xv cos uvJ0 (wun) dA, then V= _Jo (Xr) when z = and a < c, and =0 when = O and a> c. az Show also that if V= 27r- d/ e-~z sin Xv sin uvJ1 (/Az) dv, then V=J1 (Xwz) when z=0 and w < c, 8V -V=0 when z=O and z >c. az (Basset, Hydrodynamics, II. p. 33.) 22. If (1 +e2)2sin be the equation to a curve referred to oblique axes inclined to one another at an angle cot-' e, show that the equation of the curve referred to rectangular axes, with the axis of x coinciding with that of f, is co 2 y = (_)+l - J (ne) sin nx. ne 23. If 4- b4= LnJo (nx), the summation extending to all values of n given by Jo (nb)= 0, then L-n = Tb-J (nb)l.2 o (x - b4) xJ (nx) dx 8b 2J,3 (nb) - nbJ2 (nb) 32 4 - n2b2 n3 {J1 (nb)}2 b 5Jl (nb) 24. Prove that if n = k + I where k is zero or a real integer, J2 (X) + Jr J (X) is a rational integral function of x-1. For instance J2 + j2 2 J23 + J2= 2 (1+ ), and so on. o. and so on. (Lommel.) 232 EXAMPLES. 25. Prove that sin 2x J2 3J+ 5J,2 -J - 3J~ + 5J,2-.... (Lommel.) 26. Prove that if D denote d then dx Dm {t 2 Jn (yl/Z)}= (_ 2) 2/ 2 ( +m) Ja+m Dm {xzJn (Jx)} = (I )m X (-)J_ ( x). (Lommel.) 27. Prove that the equation a2 V a2 V -+ -y +V=0 ax2 ay2 is satisfied by V= (Am cos mo + Bm sin m4) (- 2p)4 dmJ (p) d (p2)m where x = p cos b, y = p sin c. Show also that {Po (~) + 2!, P () + 4! P. () + } - () () + 5! P+ (.)+... = {J0 (^ 2-)}, and obtain a corresponding expression for {JJ (/1 -_ 2)}2. 28. Show that the equation d2R 2 dR n (n + 1) dx2 X + + x2 is satisfied by either of the series (-1) x" f 1 X2 1.3... (2n + 1) 1. (2n + 3) 2 1 4x4 } 1. 2. (2n 3) (2n + 5) 4 (-1)n1.3... (2n-1) 1+ 1 X2 n- xn+ 1 t. (2n-1) 2 +. 2. (2n-1) (2n-3) 4 s t Express us as a Bessel function; and show that nUt-(n+i) = Vn'-(n+). EXAMPLES. 233 29. Verify the following solutions of differential equations by means of Bessel functions: (i) If + d2xm+ly x2x + y = 0, dXX2m1+ then 2m+1 2m y = "x 4 Cp {J-m- (2aVx/) + iJrn+ (2aplx/)}, 0 when ad, al...a2m are the roots of a2"+1l=i, and Co, C,... C2m are arbitrary constants. (ii) If d2y 2n-l dy dx2 x dx then y =x[AJn (x) + BJ_, (x)]. (iii) If __y dy x2 d2 - (2n3 - 1) x d + 2y/32 y = 0, then y _= n [AJ, (yxg) + B, (yx7)]. (iv) If 2dy dy { (, 2 r (2a - 2 + 1) X- 2Y + {a (ax 2 + lay2} y = 0 then y = _ n-a [AJ (yX) + BJ_, (yx)]. (v) Deduce from (iv) that if y -o-, o, dX2 (a form of Riccati's equation) and solve d2y x2-2 2y 0. dx2 (Lommel.) 234 EXAMPLES. 30. Prove that if u is any integral of d2u dx2 + Xu= 0, when X is a function of x, and if i =a + b, where a, b are constants, then the complete integral of d2y { a2 d + X+ [1-(n2 _ ) -2] y = 0 is y = u/f {AJ$ () + BJ_n (~I)}. (Lommel.) 31. Prove that the solution of Riccati's equation dy- ay + by2= cx dx can be made to depend upon the solution of Bessel's equation 2 d2W dw r2 d+2 r + (k2r- n2) w= 0, ~r2 Tdr where n = a/p. 32. If a bead of mass M be attached to the lowest end of a uniform flexible chain hanging vertically, then the displacement at a point of the chain distant s from the fixed end is, for the small oscillations about the vertical, (A,, cos nt + B, sin nt) V,, n where {n = {n/JMYo (nPVM) - Vmg Y, (n/3JM)} J, (np /3l - ms) - {n.IMJo (n3VM) - VmgJ, (n,8.M))} Y0 ({n/3,/ - s), u being the total mass of the chain and bead, and Pf denoting 2/N/my, where m is the mass of unit length of the chain. How are the values of n to be determined? 33. Assuming that Jo (x) vanishes when x= 2 4, show that in a V-shaped estuary 53 fathoms (10,000- 32-2 ft.) deep, which communicates with the ocean, there will be no semi-diurnal tide at about 300 miles from the end of the estuary. (See p. 113 et seq.) EXAMPLES. 230 34. The initial temperature of a homogeneous solid sphere of radius a is given by V0 = Ar-2 cos 0 (sin mr - mr cos mr): prove that at time t its temperature is m2kt U = V e-2kt provided that m is a root of the equation (ah - 2k) (ma cot ma - 1) = m22k, k, h being the internal and surface conductivities, and the surrounding medium being at zero temperature. (Weber.) 35. A spherical bell of radius c is vibrating in such a manner that the normal component of the velocity at any point of its surface is S cos kat, where S is a spherical surface harmonic of degree n, and a is the velocity of transmission of vibrations through the surrounding air. Prove that the velocity potential at any point outside the bell at a distance r from the centre, due to the disturbance propagated in the air outwards, is the real part of the expression _ _, (._~+~ /~ (ikr) s~ _ _ik(at-r+c) fn (ikr) Sd r (1 + ikc)f (ikc) - ikcf^ (ikc) ' where An (.) = (-H) x Pn dx) x P, denoting the zonal harmonic of degree n. Show that the resultant pressure of the air on the bell is zero except when n 1. A sphere is vibrating in a given manner as a rigid body about a position of equilibrium which is at a given distance from a large perfectly rigid obstacle whose surface is plane; determine the motion at any point in the air. 36. A sector of an infinitely long circular cylinder is bounded by two rigid planes inclined at an angle 2a, and is closed at one end by a flexible membrane which is forced to perform small normal oscillations, so that the velocity at any point, whose coordinates, referred to the centre as origin and the bisector of the angle of the sector as initial line, are r, 0, is qrp cospO cos nct, where pa = iTr, i being an integer and c the velocity of propagation of plane waves in air. Prove that, at time t, the velocity potential at any point (r, 0, z) of the air in the cylinder is 1 J, (n'r) 2qpap cospO Y (n_2- -- ) (na) cos nct le -k or sin kz}, " r k (nW-a2) Jp (n a) 236 EXAMPLES. where J (n'c) = 0 gives the requisite values of n', a being the radius of the cylinder, and where k is a real quantity given by the equation n'2 = n2 =k2, the upper and lower sign before k2 corresponding to the first and second term in the bracket respectively. 37. A given mass of air is at rest in a circular cylinder of radius c under the action of a constant force to the axis. Show that if the force suddenly cease to act, then the velocity function at any subsequent time varies as k Jo (k) sin kat, k2 J (kc) where a is the velocity of sound in air, the summation extends to all values of k satisfying J1 (kc) = 0, and the square of the condensation is neglected. 38. A right circular cylinder of radius a is filled with viscous liquid, which is initially at rest, and made to rotate with uniform angular velocity o about its axis. Prove that the velocity of the liquid at time t is e -,2vt J, (Xr) 2(t> XJ,' () + or, where the different values of X are the roots of the equation J, (a) = 0. Show also that if the cylinder were surrounded by viscous liquid the solution of the problem might be obtained from the definite integral | dX f e 2vXu (u) J1 (Xu) J1 (Xr) du, by properly determining < (u) so as to satisfy the boundary conditions. 39. In two-dimensional motion of a viscous fluid, symmetrical with respect to the axis r = 0, a general form of the current function is ~ =A (t+ + p4) A eat Jo (r - n where An, n are arbitrary complex quantities. (Cf. p. 116.) 40. A right circular cylindrical cavity whose radius is a is made in an infinite conductor; prove that the frequency p of the electrical oscillations about the distribution of electricity where the surface density is proportional to cos sO, is given by the equation J, (pa/v) = 0, where v is the velocity of propagation of electromagnetic action through the dielectric inside the cavity. EXAMPLES. 237 41. Prove that the current function due to a fine circular vortex, of radius c and strength m, may be expressed in the form mra e z - )" J (Xr) J (Xc) dA, the upper or lower sign being taken according as z - z' is negative or positive. 42. A magnetic pole of strength m is placed in front of an iron plate of magnetic permeability tu and thickness c: if m be the origin of rectangular coordinates x, y, and x be perpendicular and y parallel to the plate, show that Q the potential behind the plate is given by the equation m= (1 — p2) j C-_t J- (yt) dt 1-1 where p = A 43. A right circular cylinder of radius a containing air, moving forwards with velocity V at right angles to its axis, is suddenly stopped; prove that q the velocity potential inside the cylinder at a point distant r from the axis, and where the radius makes an angle 0 with the direction in which the cylinder was moving, is given by the equation = -> V cos 0 J. () cos Kat, J It (Ka) where a is the velocity of sound in air, and the summation is taken for all values of K which satisfy the equation J,' (Ka)= 0. 44. Prove that if the opening of the object-glass of the telescope in the diffraction problem considered at p. 178 above be ring-shaped the intensity of illumination produced by a single point-source at any point of the focal plane is proportional to 4 {J (z)-pJ1 (p,)} (1 -p2)2 z2 if z= 27rr/Xf, where R is the outer radius, pR the inner radius of the opening, r the distance of the point illuminated from the geometrical image of the source, andf the focal length of the object-glass. 45. Prove that the integral of the expression in the preceding example taken for a line-source involves the evaluation of an integral of the form J1 (ax) J(bx) d xJx2-2 2 238 EXAMPLES. 46. Show that J, (a) J, (bx)- anb i J 2 + b2- 2abcos) si 2nd. 2n,7r l (n-2) 1o Va + - 2cb cos s 47. Hence prove that J J (ax) J (bx) ab a sin ( J/a2 — b2- 2ab cos ) s2 J: '/2 _ e2 7r2z Jo a + b2- 2ab cos s (Struve.) 48. A solid isotropic sphere is strained symmetrically in the radial direction and is then left to perform radial oscillations: show that if u be the strain at distance r from the centre, k and n the bulk and rigidity moduli, and p the density, the equation of motion is ua21 k+ 4n a2u 4 au\ dct2 p ar2 r, with the surface condition (k + -n) a + 3k = 0. Prove that the complete solution subject to the condition stated is u = - 3 J. (1) {AP sin cpt + A cos cpt} ~+ 1 J ( ) {- (Bp sin ct+ B cos ct)} where P 2 q,2= c.2 P r2 cp being the pth root of the equation (k +4nn) aa {, J + )} 3k J (r) = 0, [~2 = c2pa2/(k + 4 n)] which holds at the surface r =a of the sphere. Show that for the motion specified BpB =p0; and [using o8, p. 53 above], prove that, if the initial values of ru, ri be 4/ (r), f (r), JO J ) ( i) (r) dr f a J (ip) k (r) dr A - A= -. cpJ{J(rqp)}I dr {J(p)2rdr fff 49. Obtain the equation of motion of a simple pendulum of variable length in the form d20 dl dO - +2 +g sin =0, EXAMPLES. 239 and show that if I = a + bt, where a and b are constants, the equation of motion for the small oscillations may be written d2U x 3iT + =0, where u = 1, x = gl/b2. Solve the equation in u by means of Bessel functions, and prove that when b/^/ga is small, we have approximately 0p(l- 4a) sin (/ —) bp (1 _ 2t) cos p and o being arbitrary constants. (See Lecornu, C. R. Jan. 15, 1894. The problem is suggested by the swaying of a heavy body let down by a crane.) 50. If the functions CQ (cos 4) are defined by the identity (1 - 2a cos + a2)-n= - (: Cn (cos 4) aS s=0 prove that Jn (/a2 + b2 - 2ab cos,) (a + 2- 2ab cos ) = 2n (n- 1), (n + s) J () n+ () (cos ). s=0 a (c (Gegenbauer.) 51. Prove that if n > m >- f00 bm (a2- U2)8-m-l J. (bx) Jn (acx) xm-+ldx b (=,b2)-matm 2n-m-111 (n - i m-1)' if a > b; and that the value of the integral is zero if a < b. (Sonine.) 52. If m> - f Jm (ax) J, (bx) J. (ex) xl-mdx [(a + b + c) (a + b- c) (b + c - ) (c +-b)] ^r. 23m-1l (m - ). ambncm provided that b + c - a, c + a - b, a + b-c are all positive; and that if this is not the case the value of the integral is zero. (Sonine.) 240 EXAMPLES. 53. Prove that J 0 2 sin(u+r) (u) du, r7r: u+ r Yrr / \ ^~cos (u + r) () du. Y (r) -2J f u w~r (Sonine and Hobson.) 54. Verify the following expansions: — oO rI (i) e C~S OJ (r sin 0) = S -. P, (cos 0), o n! (ii) Jo (r sin 0) = / 2 n P2n (cos C) J2n+ (r), J, (r sin 0) 2/2 oo (iii) s-C2SJ2S+1.(rT), (r sin 0) r- r o with the notation of example 50. (Hobson, Proc. L. M. S. xxv.) 55. Prove that /2 J1n+ (r) - J, (r sin 0) sin' 0 dO - J/ (r) / o J*-. (r sin 0) sin'+ 0 dO - ) 'V 7 Jo N Je- Yo (p) dA- -- log P p (ibid.) FORMULAE FOR CALCULATION OF THE ROOTS OF BESSEL FUNCTIONS. IN a paper, shortly to be published, "On the roots of the Bessel and certain related functions," for a MS. copy of which we are indebted to the author, Professor J. MCMahon obtains the following important results, the first of which has already been given in part. (i) The sth root, in order of magnitude, of the equation Ji () = 0 (S) m-1 4 (mr- 1) (7m - 31) ~- 8, 3 (8/3)3 32 (m- 1) (83m2 - 982m + 3779) 15 (8s)5 64 (m - 1) (6949m3 - 153855m2 + 185743m - 6277237) 105 (8,8)7 where 3 = r (2n + is- 1), m = 4n2. (ii) The sth root, in order of magnitude, of the equation J,' (x) = o (s) m+3 4 (7m2+ 82m-9) is Xl- Y Sy 3(8y)32 (83m3 + 2075n2 - 3039m + 3537) 15 (8y)3 where y= r (2n + 4s + 1), = 4n2. (iii) The sth root, in order of magnitude, of the equation d {I-$ J ()}-= 0 s) rn t 7+ 7 4 (7m2 + 154m + 95) iy 3 (8y)~ 32 (83m3 + 3535m2 + 3561mr + 6133) 15 (8)y)3 - where, as above, y,= - (2n + 4s + 1), m = 4,2. G.M. 16~~~~~~~ G. M. 16 242 FORMULZE FOR CALCULATION OF THE (iv) The sth root, in order of magnitude, of the equation (x) + (y-log 2) Jn (x)= 0 is given by the series for x(,) in (i) if 3 - w7r be therein substituted for p. (v) The sth root, in order of magnitude, of the equation d{ YE(x) + y-log 2) J ()} 0 is given by the series for x() in (ii) if y - 1 7 be therein substituted for y. [The y in the expression here differentiated is of course Euler's constant, and is not to be confounded with the y in the expression for the root.] (vi) The sth root, in order of magnitude, of the equation Gn (~)_ Gn (P)=O, p>1 G, (x) GJ (px) 0 p where - Gn (x) = Y (x) + (y- log 2) J, (x), or, which is the same, Yn (x) Y, (px) 0 Jn () J. (px) i8s x()=+p +p q- 2 r - 4pq+ 2p3 is - (S)1 + - + -...... n.,~ 8a +......, where sr M - 1 4 ( -l )(n- 25) (p3- ) where 8 P-l5 3( - 1' 8p 3(8p) (p - 1) 32 (m- 1) (mn2- 114nm + 1073) (p5- 1) =- -) 5(8p)(p-) -- m= 4) (vii) The sth root, in order of magnitude, of the equation Y' (*) Y' (P1) 0, p 1, J' (X) J' (x) is given by the same formula as in (vi), but with n m+3 4 (r2 + 46m - 63)(p - 1) P 8p ' 3 (8p) (p- 1) 32(m3 + 185m2-2053mn + 1899)(p5-1) 5 (8p)5 (p- 1) [Of course here also the G functions may be used instead of the Y functions without altering the equation.] (viii) The sth root, in order of magnitude, of the equation d d dX {x-2 Y (x)} Cd {(px)-2 Y (p)} dx- = 0,O p>l, dx- Jx ()} d {(px) J,- (p)} ROOTS OF BESSEL FUNCTIONS. 243 is also given by the formula in (vi), but with m + 7 4 (m2 270m - 199) (p3 1) P-' 8p q 3(8p)3(p-l) 13 8p 3 (sp8" (p- l) 32 (m3 + 245rn2 - 3693mn + 4471) (p5 - 1) 5 (8p) (p- 1) [As before the G functions may here replace the Y functions.] The following notes on these equations may be useful. 1. Examples of the equation in (i) are found in all kinds of physical applications, see pp. 56, 96, 178, 219, and elsewhere above. When n = the equation is equivalent to tan x = x, which occurs in many problems (see pp. 113, 191, above). The roots of this equation can therefore be calculated by the formula in (i). 2. The equation of which the roots are given in (ii) is also of great importance for physical applications, for example it gives the wave lengths of the vibrations of a fluid within a right cylindrical envelope. It expresses the condition that there is no motion of the gas across the cylindrical boundary. [See Lord Rayleigh's Theory of Sound, Vol. n. pp. 265-269.] When n =, the equation is equivalent to tan x = 2x, and when n = 3, it is equivalent to 3x tan x = 3 - 2x2' and other equivalent equations can be obtained by means of the Table on p. 42 above. 3. The roots of the equation given in (iii) are required for the problem of waves in a fluid contained within a rigid spherical envelope. The equation is the expression of the surface condition which the motion must fulfil, and x= Ka, where a is the radius. The roots therefore give the possible values of K. (See Lord Rayleigh's Theory of Sound, Vol. II., p. 231, et seq.) When n =, the equation is equivalent to tan x = x, given also by the equation in (i) when n = 3. Again when n = 3 the equation is equivalent to 2x tan x =2 -2 -_ X2 which gives the spherical nodes of a gas vibrating within a spherical envelope. 16-2 244 ROOTS OF BESSEL FUNCTIONS. 4. The roots of the equation in (vi) are required for many physical problems, for example the problem of the cooling of a body bounded by two coaxial right cylindrical surfaces, or the vibrations of an annular membrane. (See p. 99 above.) The values of x and px are those of Ka, Kb, where a, b are the internal and external radii. The roots of the equation thus give the possible values of K for the problem. 5. The roots of the equation in (vii) are required for the determination of the wave lengths of the vibrations of a fluid contained between two coaxial right cylindrical surfaces. It is the proper extension of (iii) for this annular space. As before, x and px are the values of Ka, Kb, where a, b are the internal and external radii. 6. In (viii) the equation given is derived from the conditions which must hold at the internal and external surfaces of a fluid vibrating in the space between two concentric and fixed spherical surfaces. The values of x and px are as before those of Ka, Kb, where a, b are the internal and external radii. The roots thus give the possible values of K for the problem. 7. If for low values of s the formulae for the roots are any of them not very convergent, it may be preferable to interpolate the values from Tables of the numerical values of the functions, if these are available. In conclusion, it may be stated that the ten first roots of J (x) = 0, as calculated by Dr Meissel and given in the paper referred to below, are kl = 2-40482 55577 k2 = 552007 81103 k3 = 865372 79129 k4 = 1179153 44391 k, = 14-93091 77086 k6 = 18-07106 39679 k7 =2121163 66299 k8 = 24-35247 15308 k = 27-49347 91320 kl0= 30-63460 64684 while, for larger values of n, kn = (n - I) 7r + h01 - h2r3 + h385 - h47 +h5 9 -... where = Log hi = 8-59976 01403 Log h2 = 741558 08514 Log h3 = 690532 68488 Log h4= 6-78108 01829 Log h = 6-92939 63062 and Log h, means the common logarithm of lh increased by 10. EXPLANATION OF THE TABLES. TABLE I. is a reprint of Dr Meissel's "Tafel der Bessel'schen Functionen I, und 41," originally published in the Berlin Abhandlungen for 1888. We are indebted to Dr Meissel and the Berlin Academy of Sciences for permission to include this table in the present work. The only change that has been made is to write Jo (x) and J1 (x) instead of I, and 1l. Three obvious misprints in the column of arguments have been corrected; and the value of J0(1'71) has been altered from -3932... to -3922... in accordance with a communication from Dr Meissel. Table II. is derived from an unpublished MS. very kindly placed at our disposal by its author, Dr Meissel. It gives, for positive integral values of n and x, all the values of J,,(x), from x= 1 to x =24, which are not less than 10-18. The table may be used, among other purposes, for the calculation of J, (x) when x is not integral. Thus if x lies between two consecutive integers y, y + 1 we may put x = y + h, and then Jn (x)= J (y) + hJ, ( y) + ( J." (Y) +... =, (y) + h- 1 4n (y) - ()} h2+ n (n - 1) - /\ + - ( ( 1) J (y) +Y J We take this opportunity of referring to two papers on the Bessel functions by Dr Meissel contained in the annual reports on the OberRealschule at Kiel for the years 1889-90 and 1891-2. It is there shown, among other things, that, when x is given, there is a special value of n for which the function J,, (x) changes sign for the last time from negative to positive; that the function then increases to its absolute maximum, and then diminishes as n increases, with ever increasing rapidity. Table III., which is taken from the first of the papers just referred to, gives the first 50 roots of the equation J (x) =0, with the corre 246 EXPLANATION OF THE TABLES. sponding values of J0 (x), which are, of course, maximum or minimum values of J0 (x) according as they are positive or negative. Tables IV., V., and VI. are extracted from the Reports of the British Association for the years 1889 and 1893. The Association table corresponding to V. was thought too long to reprint, so the tabular difference has been taken to be '01 instead of '001. These tables do not require any special explanation: the functions I, are the same as those denoted by that symbol in the present work. TABLE I. 247 I x Jo(x) - J1() I i I I I 0ooo00 O'OI 0'02 0o03 0'04 0'05 o-o6 0'07 o'o8 O'I0 0'II 0'12 0-13 0'I4 o0I O'I7 o I8 0'19 0'20 0'21 0'22 0'23 0o24 0.25 0o26 0'27 O'28 0'29 0'30 0-31 0'32 0'33 0'34 0O35 0o36 0'37 o'38 o'39 0'40 I I'000000000000 0'000000000000 o099997500oI56 0o999900002500 0o9997750I2656 o0999600039998 0'999375097649 0o999I00202480 0o998775375IO5 0o998400639886 0o997976024926 0o997501562066 0o996977286887 o0996403238704 0O995779460562 o'995105999233 0o994382905214 0o99361023272I 0'992788039685 o099I9I6387745 o'990995342249 0'990024972240 o'989005350457 0'987936553327 o0986818660958 0o98565I757I3I 0o984435929296 o0983171268563 0'981857869696 0'98049583I 1102 0'979085254825 0'977626246538 0o976II8915533 0'9745633747II O'972959740576 0o971308133222 0o969608676323 0o967861497I27 0'966066726439 0'964224498614 0'962334951548 O0960398226660 O'004999937500 O'4I 0-9584I4468885 0-200722502946 0-009999500008 0-42 0-95638-'826663 0-205403409-75 O-OI49983 I 2563 0-43 0-9543o6451921 0-2ioo6S948SI8 o-oi9996OOO267 0-44 0-952I825ooo67 -0-2147IS774133 -0-024992IS83I4 0-45 0-950OI2I29972 - 0'2I9352539483 - 0-223969goo- 0-029986502025 0-46 O'947795503959 370 - 0-034978566876 0-47 O'945532787790 -0-2285705I3659 0-03996SOOS5 32 - 0-2331540376i I O'4S 0-943224'5o650 - 0-044954452875 0-49 0-940869765 I 37 - 0-23772013I905 - 0-049937526036 0-50 0-938469807241 - 0-24226S457675 010549i6S54430 0-5 I 0-936024456336 - 0-246798677529 01059892o647SI 0-52 0-933533895i63 -0-25T3IO455583 o-o64862784 I 5 7 0-53 0-930998309SI2 - 0-25580.3457487 o-o6982864000I 0-54 0-928417SS97IO -0-260277350453 -- 0-26473 I S0328 I -0-07478926oi6i 0-55 0-925792827604 - 0-079744272921 0-56 0-923I233I9544 - 0-26gi66486388 - O'OS46933)07032 0-57 0-920409564868 - 0-27358IO 7I836 - o-oS963 599 I 743 0-58 0-91765I766IS7 - 0-277975233357 O' 282348646 -Si - 0-09457 I 956833 O'59 0-914850129363 j - 0-099500832639 o-6o O'9I2004863497 0'286700988o64 - O' I0442225009 I o-6i o-gogii6iSo9io -0-29103I937312 - 0-109335840739 o-62 0-905IS4297I24 - 0-29534I 1748 I I - O' I I424I236785 o-63 O'903209430845 - 0-299628383050 O' II 913807 II I3 o-64 0-90OI91803946 - 0-30' )S93246349 O'I24025977323 o-65 O'SWIP641447 - 0-1308135450885 O' I 28904589754 o-66 O'S94029I7498 - O'3I23546847'S O'I33773543525 o-67 0-890884625356 -0-3i655o637815 o' I 86324745 5 3 o-68 0-88769823737I 0-3207230020SO 0"2487I471373 - O' I43481OI9596 o-69 0-884470244964 - O' I483 I 88 I 6273 0-70 O'88I200888607 O' 328995 74 I 540 - O'I 5 3 I 45 5033099 0-71 0-877S90411804 0-33330955 I0438 -0-1579607I95i6 0-72 0-87453go6IO70 0-337 I 70477956 o-i62764IO59IS O' 7 3 0-87II470859IO -0-341220346045 oi675553036S7 0-74 o-8677I4738SOI - 0-345244SIS737 O'I72133339552I9 0-75 o-864242275i67 - 0-349243602 I 75 O'I 77099703954 o-76 o-S6072995,336i - 0-3532i64046-2 O' ISIS52I944o6 i62936538 0-77 O'S57I78034643 - 0-357 o-iS659IO72i96 0-78 0-853586783I57 - o —6io829IO50 -O' I 913 I 5 984074 0-79 o-849956465910 - 0-36497604I342 o- I 960265 7 795 5 o-8o o-846287352750 - 0-368842046094 - 0O004999937500 - 0ggg009999500008 - 0I049983 I 2563 - 0-019996000267 - 0'024992188314 - 0-029986502025 - o-034978566876 - 0039968008532 - 0'044954452875 - 0049937526036 -- o054916854430 - 0o059892o6478I - o0o64862784I57 - 0'069828640001 - 0'074789260161 - '07974427292I - 0'084693307032 - o008963599 I 743 - 009457 I 956833 - 0og099500832639 - 0'I0442225009I - 0I109335840739 - 0II4241I236785 - 0 I I913807 I I I3 - I 24025977323 - 0'I28904589754 - 0'I33773543525 - 0138632474553 - 0' I4348101I9596 - 0'I483 I 88 I 6273 - 'I 53145503099 -0-I57960719516 - 0'I62764105918 - 01I67555303687 -- 'I723339552I9 - 0 I77099703954 - 0I818521944066 - 0'I86591072i96 - 0'I913I5984074 - 0'I96026577955 I 0'40 o'4I 0'42 0'43 0'44 0'45 0'46 0'47 0'48 0'49 0'51 -5 I 0'52 0'53 0'54 0'55 0'56 0.57 o058 0.59 o-60 o-6I 0'62 0o63 0'64 0'65 o-66 o-67 o-68 o-69 0'70 0-71 0'72 0'73 0'74 0'75 0'76 0O77 0o78 o'79 o-8o x I Jo (x) o096o398226660 0o9584I4468885 0-956383826663 0-95430645 92I 0'952182500067 09500I21I29972 0'947795503959 0'945532787790 0'943224150650 0o940869765 I 37 o093846980724I 0'936024456336 0o933533895 63 0o930998309812 092841778897I0 0'925792827604 0'923I233I9544 0'920409564868 0-91765I76618 7 0'914850129363 0'912004863497 -ggI90911618090gI 0'905I84297I24 0'903209430845 0-900191803946 0'897I3I641447 O'894029171498 0-890884625356 0-88769823737I 0'884470244964 o0881200888607 0o8778904118804 0o874539061070 0o87II470859I0 0o8677I473880I 0'864242275167 o-86072995336I 0'857I78034643 0'853586783157 0o8499564659Io 0o846287352750 - 0-200722502946 - O'205 403409.3 ) 7 5 - 0-2ioo6S948SI8 -0-2147IS774133 - O'2I9352539483 - 0-223969goo)'70 -0-2285705I3659 - 0-2331540376i I - 0-23772013I905 - 0-24226S457675 - 0-246798677529 - 0-25T3IO455583 - 0-2558034571487 -0'260277350453 -- 0-26473 I S0328 I - 0-26gi66486388 - 0'273)58IO7I836 - 0'27797523")357 0 - 28 2 348646 8 I 0'286700988o64 -0-29103I937312 - 0-29534I 1748 I I - 0-299628383050 - 0-303893246349 - 0-308135450885 - O'3I23546847'S -0-3i655o637815 0-3207230020SO 0'-2487I471373 O' 328995 74 I 540 O' 3330955 I0438 0-337 I 70477956 -0-341220346045 - 0-345244SIS737 - 0-349243602 I 75 - 0-3532i64046-2 -0-357i62936538 - o —6io82qIO503) - 0-36497604I342 - 0-368842046094 - 0'200722502946 - o020540340937 5 - 0-210068948818 -0'214718774133 - 0'219352539483 - 0o223969900370 - 02285705 13659 - 0'23315403761 I - 0'23772013I905 - 0'242268457675 - 0'246798677529 - 0'25T3 I0455583 - 0'255803457487 - 0'260277350453 - 0'26473 I 803281 I - 0'269166486388 - 0'27358I07I836 - 0'277975233357 0-o28234864638I - 0'286700988o64 - 029103I937312 - 0'29534I 1748 II - 0'299628383050 - 0'303893246349 - 0'308135450885 - 0'3I2354684718 — 0-316550637815 - 0'320723002080 - 0'324871471373 - 0328995 74 I 540 - 03330955I0438 - 0337 I 170477956 - 0'341220346045 - 0'345244818737 - 0'349243602 I 75 - 0'353216404632 - 0357 162936538 - 0'361082910503 - 0-36497604I342 - 0'368842046094. _ - J1 (X) - 0I96026577955 - I I I 248 TABLE I. (continued). x2 Jo (x) - J, (x) 1T I 0o80 o0846287352750 i - 0o368842046094 o'8I 0o82 o-83 0O84 0o85 o086 0O87 o'88 o089 o'9I 0'92 o'93 o'94 o'95 o096 o097 o098 0o99 ogg I *'OO I 'OI I '02 I 03 104 105 P07o I 'o7 1 '09 -1I2 x'13 I'I4 I'I5 i-i6 1'17 I-19 I '20 0'842579716344 o0838833832154 0o835049978414 0'831228436o09 0o827369488950 0'823473423352 0'819540528409 0'8I5571095868 0'8II565420110 0-807523798123 0'803446529473 0'799333916288 0'795186263226 0O791003877452 0o786787068613 0'782536148813 0'778251432583 0'773933236862 0O76958I88O965 0o765197686558 0'760780977632 0'756332080477 0'751851323654 0'747339037965 O'742795556434 0'738221214269 0'733616348841 0o728981299655 0'724316408322 0'719622018528 0o714898476008 0'710146128520 0o705365325811I 0'700556419592 o'695719763505 o-690855713099 o0685964625798 o-681046860871 o0676Io02779403 0'671132744264 - 0-37268o644052 P2 I o-666137120OS4 0-500829672641 - O'37649,556779 1-22 o-66iii62732I4 0-503333567025 - 0-38027450S I 36 I-23 o-65607057I7o6 0-505800572628 - 0-384029224303 1-24 o-651000385275 0-508230524394 - 0-387755433798 1-25 o-6459o608527I O' 5 i o623260320 - 0-3914528675o6 I'26 o-6407SS044651 O' 5 I 29 7S621467 -0-39512I258696 1-27 o-635646637944 O'5I529645 I971 1-28 o-63048224I224 - 0-517"7659go6i - 0-39S760343044 I'29 o-6252952-'2074 0-402369858653 1) - 0-519SI89I3o63 -0-405949546079 I'30 o-620085989562 - 0-522023247415 - 0-409499 I 48347 I'31 o-614854894203 - 0-52418945868o - 0-413018410976 I'32 o-6o96O2327933 0-5263I74o6556 - 0-0650708i996 I'33 o-604328674074 0-5284o6953885 - O'4I99649I I97I 1 '.)4 O'599034317304 0-530457966666 -0-42339i654020 I'35 0-5937i9643626 -0-5324703i4o63 - 0'4267S7o63S33 I-36 0-588385040333 - 0-5344438684i8 - 0-430150899695 I-37 0-583030895983 - 0-536378505258 - 0-4334829225o6 1-38 0-577657600358 - 0-538274103303 0'4,,6782895795 I-39 0-572265544440 - O' 540 I 3054448 I 0-440050585745 I '40 0-566855I20374 0-541947713931 0-443285761209 I-4I 0-561426721439 0-543725500014 0-446488193730 I'42 0-555980742OI4 0-545463794323 O'449657657556 1-43 O'550517577543 -0-547i6249i6S6 -0-452793929666 I'44 0-5450376245IO -0-54SS2I49OI79 - O'45589678977S I'45 0'53954I280398 - 0-55044o6gi 132 - 0-45S966020374 I '46 0-53402S943664 - 0-552OI9999I33 - 0-4620OI4o67I5 1-47 0-528501013700 - 0-553559322039 - 0-465002736858 I -48 0-522957890804 - 0-555058570983 - 0-4679698oi675 I'49 0'5I7399976I46 - 0-5565 I7660374 - 0-470902394866 "50 O'5II827671736 - 0-551-9365079IO -- 0-4738003 I 2980 I'51 0-5o624I.380391 - 0-5593150345S2 -0-476663355426 "52 0-5oo64I505700 -0-56o653i64677 - O'479491324496 F5 3 0'49502845I994 - o,56I9508257S6 - 0-4822S4025373 I-54 0-489402624312 - 0-563207948So6 - O'4S5041266I54 1-55 0-483764428365 - 0-564424467949 - 0-487762857858 1-56 0-478,14270507 - 0-565600320742 - 0-4go44S614448 I-57 0-472452557702 — 0'566735448033 - 0-49309835284 I I-58 0-466779697485 - 0-567829793994 - 0'49571 i892924 1-59 0-46iog6o97935 - 0-5688833o6l26 - 0-498289057567 i -6o 0-455402i67639 O'569895935262 I- I - 0o372680644052 - 0376491556779 - 0380274508 I 36 - 0o384029224303 - 0o387755433798 - 0391452867506 - 039512I258696 - 0-398760343044 - 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J1 (x) - 0.498289057567 - 0o50082967264I - 0'503333567025 - o0505800572628 - 0'508230524394 - 0'510623260320 - o0512978621467 - 0o51529645 971 - 0o51757659906I - 0o5198I89I3063 - 0522023247415 - 0524189458680 - 0-526317406556 - 0o528406953885 - 0o530457966666 - o5324703I4063 - 0.534443868418 - 0'536378505258 - 0.538274103303 - 0540130544481 - 0'541947713931 - 0543725500014 - o0545463794323 - 0547I62491686 - o548821490179 - 0o55044o691132 - 05520I9999I33 - 0o553559322039 - 0555058570983 - 0.556517660374 - 055-79365079Io - 0559315034582 -o 560653164677 - 0o56I950825786 - o0563207948806 - 0o564424467949 - 0565600320742 - 0'566735448033 - 0567829793994 - 0568883306126 - o0569895935262 I I I TABLE I. (continued). 249 x Jo, ((x) J x J-o(x) -J1(x) 1~~~~0z ) -~ x I'60 I 6I I163 i -63 I -64 I -65 I 66 I167 I 68 I '69 I170 1'71 I 72 1'73 1'74 I'75 1I76 1'77 1-78 I'79 I 80 iS8i I 82 I83 1'84 1I85 I86 r87 I 88 I 89 1'90 I go I'-9 I 1-92 '93 '94 I-95 I 96 1'97 I 98 1I99 2'00 0'455402167639 0'449698315660 0'443984951500 0'438262485071 0-432531326660 0-426791886896 0-421044576715 0-415289807326 0-409527990I83 0'403759536945 0'397984859446 0-392204369660 0-386418479668 0o380627601627 0'374832147732 o0369032530185 0'363229161163 0-357422452782 0-351612817064 0-345800665906 0-339986411043 0'334170464016 0'328353236143 0-322535138478 0'316716581784 0'310897976496 0-305079732690 0-299262260050 0'293445967833 0o286631264839 0'281818559374 0'276008259222 0-270200771606 0-264396503162 0o258595859901 0'252799247180 0'247007069667 0-241219731308 0'235437635298 0o229661184046 0-223890779141 -0569895935262 - 0570867635566 -0'571798364542 - 0572688083032 -0'573536755217 - 0'574344348624 - 057511 0834122 -0'575836185927 -0'576520381599 - 0577163402048 - 0577765231529 -- 578325857645 - 0578845271345 - 0579323466925 -0'579760442028 - 0580156197639 - o580510738087 - 0580824071043 - 0'581096207515 -0'581327161851 - 0581516951731 -0-581665598167 - 0581773125501 - 0581839561397 - 0581864936842 - 0-581849286141 - 0581792646910 - 0581695060074 - 0581556569863 - 0581377223803 -0-581157072713 -0-580896170703 - 0580594575158 - 0580252346743 - 0579869549389 - 0'579446250290 - 0578982519892 - 57847843 1892 0577934063221 o- 0577349494047 - 0576724807757 2'00 2'01 2'02 2-03 2'04 2'05 2-06 2'07 2-08 2-09 2-10 2'11 2'12 2'13 2'14 2'15 2'16 2'17 2'18 2-19 2'20 2'21 2'22 2'23 2'24 2'25 2'26 2'27 2'28 2-29 2'30 2'31 3'32 2'33 2'34 2'35 2-36 2'37 2-38 2'39 2'40 0'223890779141 0'218126821326 0-212369710458 0-206619845483 0'200877624399 0'I95143444226 0'189417700977 0-I8370078962I O'177993104055 0-172295037073 o0166606980332 O'I60929324324 0'155262458341 0'149606770449 O'143962647452 0-138330474865 O'13271063688I 0-127I03516344 O'I21509494713 O'115928952037 O'110362266922 0O104809816503 00o99271976413 O'093749120752 0-088241622061 0-082749851289 '077274177765 0'071814969172 o-o66372591512 0o060947409082 0'055539784446 0-050150078400 0-044778649952 0-039425856288 '034092052749 0-028777592796 0'023482827990 0-OI8208107961 0I012953780380 0-007720190934 0-002507683297 - '576724807757 - 0576060090955 - 0575355433450 - 05746I0928248 - 0573826671543 - 0573002762707 - 0572139304279 - 0571236401957 - 05 70294164587 - 0569312704151 - 0568292135757 - 0567232577628 - 05663415 I091 - 0564996980564 - 0563821193544 - 0562606920596 -0-561354295339 - 0560063454436 - 0558734537577 - 0557367687469 - 0555963049819 - 0554520773326 -- 55304Io09659 - 551523913451 - 0549969642278 0'548378356647 - 0546750219981 - 0545085398603 - 0543384061721 - 54I6463814I2 - 0'539872532604 - o538062693065 - 0536217043381 -0534335766941 - 053241904992I - 0'530467081267 -0'528480052675 - 0526458158577 -0-524401596119 - 0'522310565146 - 0520185268182 250 TABLE I. (continued). - x Jo (X) - J1 (X) Jo () - J1 (X) I I 2'40 2'41 2'42 2'43 2'44 2'45 2'46 2'47 2'48 2'49 2'50 2'5 I 2'52 2'53 2'54 2'55 2'56 2'57 2'58 2'59 2'60 2'61 2'62 2'63 2'64 2'65 2'66 2'67 2'68 2-69 2'70 2'7I 2'72 2'73 2'74 2'75 2'76 2'77 2'78 2'79 2-80 + o'oo2507683297 - o00oo2683400894 - 0oo007852722067 - 0'012999942745 - 0'018124727564 - 0'023226743305 — o0028305658919 - 0'033361145552 - 0038392876569 - 0'043400527581 - 0'048383776468 - 0o053342303407 - 0oo58275790893 - 0o063 183923765 -0 o068066389230 - 0072922876886 - 0'077753078750 - 0082556689272 - 0087333405369 - 0092082926441 - o096804954397 - 'I101499I93675 - 0I06165351268 O 0I Io80313674I - 0I 15412262258 — O' I I9992442602 - 0'I24543395193 - O'I29064840 I15 - 0I 33556500 I 33 - 'I380I8100713 - O'142449370046 - 0'146850039066 - 01I512I984I469 — 0155558513735 - O' I 59865795147 - 0'I64141427809 - -0168385 156663 - 'I 1725967295 15 -0I 76775897046 - 0IS809224I2832 -O' I85036033364 11 - 0520I85268182 -- 05I80259I04I3 - 05I5832699667 - 0o5 13605846395 - 05 I 34556365I - 0'509052067073 - 0506725574866 - 0'504366307779 - 0501974489084 - 0'499550344558 - 0497094102464 - 0'494605993526 - 0492086250909 - 0'489535 I 0203 - 0'486952809393 - 0'484339588844 - 0o48I69569I279 - 047902I36I753 - 04763 I6847635 - 0'473582398581 - 0470818266518 - 0468024705615 - 0'46520I972264 - 0'462350325057 - 0'459470024758 - 0-45656I334286 - 0'4536245 I8688 - 0'450659845 I 15 - 0'447667582797 - 0'444648003025 - 0'4416013791 8 - 0o438527986406 - 0'435428102199 - 0432302005768 - 0'429149978317 - 0'425972302958 - 0o422769264686 - 0'419541150353 - 0'46288248646 - 041I3010850055 - 0'409709246852 2'80 2'8I 2'82 2-83 2'84 2'85 2-86 2'87 2'88 2'89 2'90 2'91 2'92 2'93 2'94 2'95 2'96 2'97 2'98 2'99 3'00 3'0I 3'02 3'03 3'04 3'05 3'06 3'07 3'08 3'09 3'I0 3'12 I 31I2 3'13 3'I4 3'15 3'i6 3'17 3'I9 3'20 I - 0185036033364 - oI89II65I8066 - 'I93163629309 - 0'I97177132431I - 0'20115679575I - 0o205I02390590 - 032090I369I285 - 0'2 12890475203 - 0'21673252276I -- 0'220539617438 - 0'2243 1I 545792 - 0'228048097475 - 023 I 749065248 - 0'235414244994 - 0'239043435734 - 0'242636439638 - 0'246193062043 - 0'249713111 I I464 -0'253196399605 - 0256642741376 - 0'26005 I954902 - 0'263423861537 - 0-266758285876 - 0'270055055766 - 02733 I14002318 - 0'276534959916 - 027971776623I - 0'282862262330 - 0285968292I86 - o0289035703688 - 029206434765 I - 0'295054078324 - 0'298004753302 - 0'300916233531 - 0o303788383321 - 0306621070350 - 0'309414165674 - 3 I 2 I 67543732 -0'314881082360 - 3I7554662788 -0'320I88169657 I 0-409709246852 -0-4o6383733o66 0-403034604450 O' 399662 I 58463 -0-396266694238 - 0-392S485 I2558 - O' 3894079 I 5 829 - 0-38594520805 I - 0-38246o694795 - O'3 78954683 I 74 - O'37542748 I 8 I 3 - 0-371879400828 - O' 3683 I 075 I 792 0-36472IS47712 O-36i I I 1300300 I O'3 5 7 4845 3 3446 0-353836756i87 O'350i69989683 -0-346484553686 - 0-3427SO7692i6 -0'339058958526 - O'3353194450S I - 0"')3I562553524 - 0-3277886o965 I - O'32399794033SO - 0-320190873724 -0-3i6367738762 - O'3I252SS656og - 0'3oS674585389 - 0-304SO5230202 -0-300921133IOI - 0-297022628058 - 0-293 I 10049938 - 0-28gi83734465 - 0-285244018200 - 0-28 I 291238504 - O'2773257335 I4 - O'273347842I IO - O'269357903890 - O'265356259I34 - O'26 I 343248781 - 0409709246852 - 0'406383733066 - 0'403034604450 - 0399662 I 58463 - 0'396266694238 - 0'392848512558 - 0'389407915829 - 0o38594520805 I - 0'382460694795 - 0'378954683174 - 0'37542748 1 8 I 3 - 0-371879400828 - 0'3683 I 075 1792 -- 0'36472I847712 - 0361 I I1300300I - 0'357484533446 - 0'353836756187 - 0350I69989683 - 0o346484553686 - 0O342780769216 - 0'339058958526 - 0'335319445081 - 0'331562553524 - 0'32778860965I - 0'323997940380 - 0'320190873724 — 0'3i6367738762 - 0'312528865609 - 0'308674585389 - 0'304805230202 - 0300921133101 - 0.297022628058 - 0o293I 0049938 - 0o289I83734465 - 0'285244018200 - 0'281291238504 - 0'2773257335 14 - 0'2733478421 I0 - o'269357903890 - 0'265356259I34 - o'261343248781 I I I it I I TABLE I. (continued). 251 x 3.20 3'21 3'22 3'23 3'24 3'25 3'26 3'27 3.28 3.29 3'30 3.3'I 3'32 3'33 3'34 3'35 3'36 3'37 3'38 3'39 3'40 3'4I 3'42 3'43 3'44 3'45 3'46 3'47 3'48 3'49 3'50 3'5I 3'52 3'53 3'54 3'55 3'56 3'57 3'58 3'59 3'60 I IT I Jo(X) -o' 320I88169657 - 0o32278I49I1I7 - 0'3253345I8339 o0327847I46516 - 0'330319273873 - 0.332750802 I 7 I - 0'335141636607 -0'337491685828 - o0339800861926 - 0'342069080449 - 0344296260399 - 0'346482324240 - 0-348627I97900 - 0o35073081077I - 0'352793095716 - 0'354813989067 - 0'35679343063I - 0'358731363688 - 0'360627734994 - 036248249478I - 0'364295596762 - 0'366066998124 - 0'367796659535 - 0'369484545139 - 0.37 I 30622559 - O'372734862895 - 0'374297240720 - 0'375817734085 - 0'3772963245 I I - 0o378732996992 - 0'380127739987 - 0'381480545425 - 0'382791408696 - 0'384060328649 - 0385287307591 - 0-38647235 I 282 - 0'387615468930 - 0'388716673186 - 0'389775980I44 - O'390793409330 - 0'39176898370I - J, (X) x Jo (X) - J, (X) -0-26I343248781 -,-6o -0-119I76S983701 -0'095465547I78 - O'2573 I 9214392 3-6i -0-392702729637 -0-091284136789 - 0-25328449SI29 3-62 -0-393594676939 - O-OS7 I05S77039 - O'2492394427 I 9 3-63 - O' 394444858S I 7 - O-OS29-'l I08843 - 0-245 I 84391424 3-64 -0-3952533IISSS -0-07876o'72463 -0-24Iii9688o15 3-65 - 0-39602007617I -0-074593407483 O'23704567674I 3-66 - 0-396745195072 - 0-07043 I 152776 0-232962702298 3-67 - 0-3974287 I 53S8 - o-o66273746480 0-22S87I I09797 — 68 - O' 9SO7o6S7288 -o-o62I2I 25964 0 3 5 O'224771244740 3-69 -0-39867Ii64315 0-057974827SO2 -0-22o663452985 3-70 - 0-399230203371 0-053833987745 0-2i6548080719 3-71 - 0-399747864713 0-04969934o694 0'2 I 2425 474424 3'72 - O'400224211942 0'04557I22o667 0-208295980854 3'73 0-40o65931 I 994 0-041449960775 - 0-204I59946997 3-74 - 0-401053235132 -0-037335893I93 - 0-2000I 7720051 3-75 - 0-40IV6054936 - 0-033229349130 - O' I 95869647392 3-76 - 0-40I7 I 7S48294 - 0-029 I 3o658803 - O' 1917 i 60765 43 3-77 - 0-40I988695389 - 0-02504OI51411 -0-IS7557355I45 3-78 - 0-4022iS679692 - 0-020958 I55IO2 -0-IS3393S30929 3-79 - 0-402407887951 - o-o i 6SS4996950 -O'I7922585i682 3-SO - 0-402556410179 - 0-012821002927 O'I750537652i8 YS I - 0-402664339640 - 0-008766497873 O' I 70877919353 YS2 - 0-402731772S45 -0-00472IS05471 -o-i6669866iS69 3-S3 - 0-402758809533 - o-ooo68724S221 - o- i 625 i 6340485 3-84 - 0-402745552664 + 0-003-036852592 -0-15833130283I 3-S5 - 0-402692io8403 +0'007350176giS - O' 154I43896414 3-86 - 0-4025985S6i io + O-OI 135240597 5 0-149954468592 -22227,Z 3-87 -0-402465098327 + O-OI53U O'145763366540 3-SS - 0-40229 I 76076i +O'OI93223o9635 O'I4I57093722 I 3-89 - 0-4020786922SO + 0-023289353237 -O'I37377527362 3-90 - O'4Oi826ol4SSS +0-027244039621 - 0'1331834834i6 3-91 - 0-4015338537I9 +0-03HS6056727 -0-128989IP538 3-92 - 0-401202337020 +0'0351150939IS - O'124794877553 3-93 - 0-400831596137 + 0-03903o842oo6 0 - 12o6o i oo692 7 3-94 0-40042 I 765 502 +0-042932993278 o' i i 6407SS4739 3-95 - 0-399972982615 +0'046S2124152I O' I I 2215855647 3-96 0-399485388031 + 0-05o695282047 -0-10SO25263865 3-97 - 0-398959I25344 + 0-05455481 I 719 -0-IO3836453128 3-98 - 0-39839434I I72 + 0-05S39952S975 - o-o99649766668 3-99 0-39779 I I S5 I 39 +o-o62229I33855 - 0-095465547178 4-00 O' "97149So9864 + o-o66043328024 - J1(x) - 0o26I34324878I - 0'2573I9214392 - 0253284498129 - 02492394427 I 9 - 0'245I84391424 - 024 I Ii96880I5 - '23704567674I - 0'232962702298 - 0-22887I I09797 - '224771244740 - 0o220663452985 - 0'216548080719 - 0'212425474424 - o0208295980854 - 0204I59946997 - 0'2000I772005I - 0I95869647392 - oI91716076543 - 0-I87557355I45 - 0-I83393830929 - 0 I 79225851682 - 0'I75053765218 - 0 I 70877919353 - 0'I66698661869 - 0I-62516340485 - 0'I5833130283I - 0'I54I438964I4 - o0 149954468592 - 0I145763366540 - 0oI4I57093722I - 0'I37377527362 - 0'I33183483416 - 0-I28989I51538 - OI124794877553 - 0I120601006927 - O-I I6407884739 - -I I2215855647 - -0Io8025263865 - 0I03836453128 - o-o99649766668 - o0095465547 178 I I I I x 3.60 3.6i 3.62 3'63 3'64 3'65 3-66 3'67 3.68 3'69 3'70 3'7I 3'72 3'73 3'74 3'75 3'76 3'77 3'78 3'79 3-80 3.8I 3-82 3'83 3'84 3'85 3'86 3'87 3-88 3'89 3'90 3'9I 3'92 3'93 3'94 3'95 3'96 3'97 3'98 3'99 4.00 J0(z) - '391 768983701 - 0'392702729637 - 0393594676939 - '394444858817 - 03952533 II888 - 0'39602007617I - 0396745195072 - 0o397428715388 - 0398070687288 - 0'398671I64315 - 0'39923020337I - 0'399747864713 - 0'4002242I I942 - 04006593II994 - 0'401053235 I 132 - 0401406054936 - 0'407 I 7848294 - 0'40I988695389 - 0-4022I8679692 - 0402407887951 - 0402556410179 - 0'402664339640 - 0'402731772845 - 0402758809533 - 0'402745552664 - 0402692108403 - 0'4025985861 IO - 0'402465098327 -0'402291760761 - 0'402078692280 - 0'401826014888 - 0'401533853719 - 0401202337020 - 0400831596137 - 0'400421765502 - 0'399972982615 - 0o39948538803I - 0'398959125344 - 0'398394341I72 - 0'39779 I I 85139 - 0'397 149809864 - J1 (x) - 09og5465547I78 - 0o091284136789 - 0087Io05877039 - o-o8293I1o8843 -o'o78760I72463 - 0074593407483 - 0'070431152776 - 0o-o066273746480 - o-o62I2I525964 - 0'057974827802 - 0053833987745 - 0'049699340694 - 0-04557I22o667 - 0'041449960775 -o 0037335893I93 - 0'033229349130 - 0o029I3o6588o3 - 0'02504015141 I - 0-020958155102 - 0-016884996950 -0-012821002927 - 0oo008766497873 - 0-oo00472180547 I - 0-00068724822I + 0-oo003336852592 +0'0073501769 8 + O-OI I 352405975 +0015343222272 + O'I9322309635 + o0023289353237 +0'027244039621 + 0031I86056727 +0'035 5093918 + 0'039030842006 + 0'042932993278 +0'04682124152I + 0-'050695282047 + 0'05455481 II719 + 0'058399528975 +0'062229I33855 + 0o066043328024 I I tl I I 252 TABLE I. (continued). X. J -Jo X) - J, ): 4'00 - 0397 I49809864 +o'o66043328024 4'01 4-02 4'03 4'04 4-05 4'06 4'07 4'o7 4.08 4'09 4-11 4'12 4'13 4'15 4'16 4'17 4'19 4.20 4.21 4.22 4'23 4'24 4'25 4.26 4'27 4.28 4'29 4'30 4'31 4'32 4'33 4'34 4'35 4'36 4'37 4-38 4'39 4'40 - 0'396470370937 - 0395753026909 - 0394997939273 -- 394205272445 - 0393375193748 - 0392507873396 - 0391603484474 - 0390662202921 - 03896842075I I - 0388669679836 - 0387618804284 - 0386531768024 - 0385408760984 - 0384249975834 - 0383055607963 - 0381825855461 - 0-380560g919IO - 03792610o2313 - 037792631 172 - 037655 7054368 - 0375153443190 - 0373715691507 - 03722440I574I - 0370738634848 - O369199770300 - 0367627646055 - 0366022488543 - 0364384526637 - 0362713991635 - 0360I I I7237 - 0359276139517 - 0357509296907 -0-355710830I68 - 0353880982370 - 0352019998867 - 0350128127272 - 03482056I7435 - 0346252721418 - 0344269693470 - 0-342256790004 + o-69841814795 + 0073624299 I 58 +0'077390487802 + 0o08 1140089137 + 0084872813321 +0o088588372282 + 0092286479742 + 0095966851242 + 0099629204162 +0 I03273257747 +O-106898733130 +- I I0505353352 + O I 4092843385 +OII 7660930159 +0 121209342578 +O'1247378 I545 + OI28246069984 +01I31733852860 +O'I35200897203 + ' 138646942126 +0-142071728849 +0O145475000717 + '148856503224 + 0152215984028 + -155553192978 + 0158867882130 + -162159805765 + -I65428720414 + - 68674384873 +- 17I1896560222 + 0175095009847 + 0178269499458 +O'I81419797I04 +- I84545673196 + 0I87646900522 +' I90723254265 + O'I93774512024 + OI96800453825 + '199800862145 + 0202775521923 x 4'40 4'41 4'42 4'43 4'44 4'45 4'46 4'47 4'48 4'49 4'50 4'51 4'52 4-53 4'54 4'55 4'56 4'58 4-59 4-60 4'61 4-62 4-63 4-64 4-65 4-66 4-67 4-68 4'69 4'70 4'7I 4,71 4'72 4'73 4'74 4'75 4-76 4'77 4-78 4'79 4'80 J0 (x) - 0342256790004 - 0-340214269569 - 0338142392830 - o0336041422538 - o3339 1623508 - 0331753262593 - 0329566608658 - 0327351932553 - 0325109507090 - 0322839607016 - 0-320542508985 - 0318218491534 - 0315867835056 -0-313490821772 - 0-3I 087735706 - 0-308658862659 - 0-306204490I79 - 0'303724907535 - 0-301220405692 -- 0298691277281 — o296137816574 - 0293560319453 - 0-290959083385 - 0288334407392 - 0-285686592028 - 0283015939344 - 0'280322752864 - 0277607337557 - 0-274869999807 - 0272I I 1047384 - 0o269330789420 - 0'266529536373 - 0-263707600004 - 0260865293347 - 0-258002930679 - 025512082749I -0-252219300460 -0-249298667418 - 0246359247327 - 0o24340I360242 - 0-240425327291 - J1 (x) +0'202775521923 + 0205724220583 + 0208646748043 +0-211542896739 + 0-214412461634 + 0217255240239 + 0220071032626 + 0222859641442 + 0-225620871929 +0'228354531934 + 023I060431923 + 0233738385002 + 0236388206923 + o0239009716I03 + 0241602733636 + 0244167083306 + 0246702591599 + 0249209087721 + 0251686403603 +o0254134373919 +0o256552836097 + 0258941630330 + o261300599586 + 0263629589622 + 0265928448996 + 0268197029073 + 0-27043518404I + 0272642770917 + 0274819649559 + 0276965682678 + 0279080735843 + o28II64677493 +0-283217378945 + 0285238714404 + 0287228560970 + o289186798647 + 029I I I33IO352 + 0293007981919 + 0-294870702112 + 029670I362626 + 0-298499858I00 TABLE I. (continued). 253 x Jo(x) -J,(x) 1 4'80 - 0240425327291 ++0'298499858100 4'81 4'82 4'83 4-84 4-85 4-86 4'87 4'88 4'89 4'90 4'91 4'92 4'93 4'94 4'95 4'96 4'97 4'98 4'99 5'00 501I - 0'237431470639 - 0234420113459 - 0231391579906 - 0228346195084 - 0225284285019 - 0222206176625 - 0-219112197679 - 0216002676790 - 0212877943365 - 0209738327585 - 0206584160372 - 0203415773359 - 0200233498860 - I97037669840 - 0193828619886 - o190606683176 - 0187372194447 - 0I 84125488969 - o180866902512 - 0I 77596771314 - 0174315432057 + 0300266086117 +0-301999947217 +0'303701344899 + 0305370185627 +0o307006378837 + 0308609836942 +0-310180475336 +o 3II718212399 + 0-313222969504 +0o314694671015 + 0316133244299 +0'317538619723 + 0318910730662 +0'320249513497 +0-321554907624 + 0322826855452 + 0-324065302408 + 0325270I96936 + 0326441490501 + 032757913759I + 0328683095718 +0'3297533254I5 + 0330789790243 +0'331792456787 + 0332761294658 + 0333696276491 + 0334597377947 + 0'335464577712 +0336297857492 + 0337097202018 + 0337862599041 + 0338594039331 + 0339291516672 +0'339955027866 +o 340584572725 +o'34180154069 +0'341741777728 +o 342269452530 +0-342763190303 + 0343223005872 x 5'20 5'21 5'22 5'23 5'24 5'25 5'26 5-27 5'28 5'29 5'30 5'31 5'32 5'33 5'34 5'35 5'36 5'37 5-38 5'39 5'40 5'41 5-42 5'43 5'44 5'45 5'46 5'47 5'48 5'49 5'50 5'51 5'52 5'53 5'54 5'55 5'56 5'57 5'58 5'59 5'60 Jo(x) - 1(X) - 0110290439791 + 0343223005872 - O- o6856051931 +0o34364891705I - oI03417574396 + 034404094464I - 0099975345904 + 0344399112424 -oog96529704924 +0344723447160 - o093080989639 +0'345013978579 - 0089629537922 +0'345270739379 -o0o86175687302 +0-345493765217 -o0082719774939 -+0345683094703 - 0079262137591 +0-345838769398 - 0-075803I 11586 + 034596083380I - 0072343032791 + 0346049335349 - 0068882236587 +0'346104324405 -0065421057834 + 0346125854251 -0o061959830846 + 03461I3981085 - 0-058498889359 + 0346068764007 - 0055038566506 +0'345990265014 - 0-051579194783 + 0345878548995 - 0-048121106024 +0'345733683714 - 0-044664631371 + 0345555739809 - 0-041210101245 +0-345344790780 -0037757845318 +0'345100912978 - 0034308192484 + 0344824185600 - 0030861470832 +0'344514690673 -0-027418007614 +0'344172513049 -0-023978129221 + 0343797740393 -0-020542161155 +0'343390463171 - 0017110427996 + 0342950774642 - 0013683253380 +0'342478770844 - 0010260959967 + 0341974550584 - 0-006843869418 +0-341438215429 - 0-003432302361 + 0340869869689 - 0'000026578369 + 0340269620408 + 0003372984068 + 0339637577354 +00oo6766067573 +0'338973853000 +0OI01052355907 +0'338278562520 +0'013531533995 +0'337551823766 + 0016903287956 + 0336793757265 +0'020267305125 + 0336004486197 +0'023623274084 + 0335184136388 + 0-26970884685 +0'334332836291 5-02 -0-I71023221828 5'03 - o'67720478o98 5-04 - 0164407538685 5'05 5-o6 5'07 5'o8 5'09 5-I20 5'11 5'12 5'13 5I14 5'15 5'-1 5-I9 5-20 -o-I6I084741725 - 0-157752425645 - O'I54410929130 — 0'15060591092 -0-147701750643 - 0'144334747061 - '0140959919761 - 01 37577608269 -O0I34188152185 - 0130791891157 - '0127389164849 -01I23980312914 - 0I20565674960 - 0' 17145590523 - O 113720399033 - 0-I 0290439791 254 TABLE I. (continued). x Jo (X) -J (x) x Z Jo(x) -J(x) I~~~~~~~ ~x I 5-60 5-6i 5-62 5-63 5'64 5-65 5-66 5-67 5-68 5-69 5-70 5'7I 5-72 5-73 5-74 5'75 5.76 5-77 5-78 5.79 5'8o 5-82 5-83 5-84 5-85 5-86 5-87 5.88 5-89 5'90 5'92 5'94 5-95 5'96 5'97 5'98 5;99 6-oo +0-026970884685 +o-030309828079 + 0033639796739 + -036960484490 + 0-040271586530 +0'043572799459 + 0-046863821304 +0'050144351544 +0 05341409II35 +0-056672742533 +0o059920009724 + 0-063155598244 +0o-o066379215205 + 0-06959056932I +0-072789370930 +0'075975332017 +0'079148166242 +0-08230758896I +0'085453317250 +0-088585069926 +0-091702567575 +0o09480553257I +0-097893689100 +0-100966763183 +0-104024482698 +0'I07066577404 + 0 I I0092778957 ++011310282094I +0-11609643888I +0-119073370272 + -I12033354593 + 0-124976133333 +0-1279014500I I +0'I30809050195 +0I33698681524 + 0-136570093728 + +0I39423038646 +0O-42257270250 +0O-4507254466I + -I47868620168 +0'I5064525725I +O 33433283629I + 0333450716975 +0-332537912108 +0'33I594557948 +0-330620793320 +0-329616759609 + 0-328582600738 +0'327518463159 + 0326424495830 +0'325300850207 + 0-324I47680223 +0-322965142271 + 0-321753395193 +0'320512600255 +0-319242921139 + 0317944523919 +0-3i6617577048 +0-315262251336 +0-313878719939 +0-3I2467158333 +0'311027744304 +0-309560657922 +0 308066081529 +0'306544199716 + 0304995199305 +0o303419269333 +0301816601028 +0o300187387793 +0'298531825 85 + 0296850110895 + 0295142444729 + 0-293409028587 +0'291650066443 + 0-289865764324 + 0288056330291 + 0-286221974417 +0'284362908764 +0o282479347366 + 02805715o6204 +0'278639603186 + 0276683858128 6-oo 6-oi 6-02 6-03 6-04 6-05 6-o6 6-07 6-o8 6-og09 6-10 6-ii 6-12 6-I3 6-I4 6-15 6-i6 6-17 6-i8 6-19 6-20 6-21 6-22 6-23 6-24 6-25 6-26 6-27 6-28 6-29 6-30 6'3I 6-32 6-33 6-34 6-35 6-36 6-37 6-38 6-39 +0150645257251 + 0-153402218596 +0-156139269116 + 'I58856I75969 +0-161552708575 + 0-164228638636 +0o166883740153 + 0- 169517789443 +O'172130565159 + 'I74721848302 + 'I77291422243 + 0-179839072737 + 0-182364587942 + -I84867758430 + o187348377209 + 0189806239737 + -I 92241143934 + 0-'I9465289020I +o' 97041281434 + O-I99406123040 + -201747222949 + 0204064391629 + 0'206357442103 + o208626189957 + 0'2I0870453362 +o 213090053077 +0-215284812471 + 0-21745455753I +0-219599116876 +0-221718321770 + 0-223812006132 +0-225880006549 + 0227922162289 + 0-229938315309 + 0-231928310269 + 0233891994542 +0-235829218223 +0'23773983414I +0-239623697870 + 0-241480667734 + 0276683858128 +0o274704492725 + 0272701730538 + 0-270675796964 +0-268626919220 + 0266555326316 +0'264461249036 + 0262344919911 + 0260206573201 +0-258046444869 +0-255864772558 + 025366179557I +0'251437754842 +0-249192892918 F 0'246927453930 + 0o24464i683576' +0'242335829091 + 0-240010139225 +o-237664864220 +0-235300255786 +0'232916567073 + 0230514052652 + 0-228092968487 +0o225653571908 +0-223196121594 +0o220720877539 + 0-218228101034 + 0-215718054638 +0-213191002155 +0'21o647208606 +0-208086940207 +0-205510464342 +0-202918049537 +0-200309965435 +O'197686482769 + -195047873339 + O-I92394409984 + 0-189726366557 + -187044017898 + 01 84347639808 + o-81637509024 6-40 +o0243310604823 I TABLE I. (continued). 255 i x Jo(x) - J(x) x Jo(z) - J(x) 6'40 6'41 6-42 6-43 6'44 6'45 6-46 6'47 6-48 6'49 6'50 6'51 6-51 6'52 6'53 6'54 6'55 6-56 6'57 6-58 6-59 6-60 6-6i 6-62 6-63 6-64 6-65 6-66 6-67 6-68 6-69 6'70 6-71 6-72 6-73 6.74 6-75 6-76 6'77 6'78 6-79 6-8o + 0'243310604823 +0'245113372998 + 0246888838899 + 0248636871957 +0'250357344403 + 0252050131270 + 0253715110409 +o'255352I6249I +0'25696II77IO5 +0-258542022319 + 0260094605582 +0'261618812832 +0'263114538957 + 0264581681702 + 0266020I41682 + 0267429822386 + 02688I063018I +0'270I624743I8 +0-271485266933 + 0272778923059 + 0274043360624 + 0275278500456 + 0276484266288 +0-277660584760 +0-278807385424 + 0279924600745 +0'28IOI2166103 + 0282070019798 + 0283098103049 + 0284096359998 + o0285064737711 + 0286003186176 +0'286911658311 + 0287790109957 + 0288638499883 +0-289456789785 +o 290244944284 + 0-291002930929 + 0291730720194 + 0292428285479 +0'293095603104 +0'181637509024 +01I78913903193 + 'I76177100845 + 0'173427381364 + I170665024967 + oI67890312675 +0o165103526284 +0O162304948344 + I59494862126 + 1I56673551601 + 0153841301410 + 0I50998396839 + 0148145123790 +O'I45281768758 + I142408618801 +0o139525961513 + I 36634085000 + 013373327785 +o0130823829111 +0127906028255 + 0124980165161 +01I22046530081 +0'I191I5413617 +0-I16157106694 +0II3201900529 + I 10240086609 +01OI7271956661 + 0104297802626 +0-101317916630 + 0098332590962 + o095342 118041 + 0'092346790394 +o-08934690o625 +o o8634274139I + 0083334605375 +0'080322785255 +0'077307573684 + o074289263257 + 0071268146488 +o-o68244515780 + 0-65218663402 6-80 6-8I 6-82 6'83 6-84 6-85 6-86 6-87 6-88 6-89 6-90 6-9I 6-92 6'93 6'94 6-95 6-96 6-97 6-98 6-99 7-00 7'0I 7'02 7'03 7'04 7'05 7-o6 7'07 7-08 7-09 7'10 7'12 7'13 7'14 7I15 7-16 7-19 7-20 + 0293095603I04 + 0293732652315 + 0294339415275 + 0294915877066 + 0295462025686 + 0295977852047 + 0296463349971 + 0296918516185 + 0o297343350324 + 0297737854921 +0'298102035405 + 0298435900099 +0-298739460212 +0'299012729839 + 0'299255725950 + 0299468468391 +0-299650979874 +0-299803285973 +0'299925415120 + 03000I7398594 + 0300079270520 + 0-300111 67856 + 0300112830394 + 0300084600744 + 0300026424335 +0-299938349401 +0-299820426973 +0'299672710878 + 0299495257720 +0'299288 126879 + 0-299051380502 +0-298785083486 +0-298489303478 + 0'29816411086I + 0297809578741 + 0297425782943 + 0'297012801997 +0o296570717126 + 0296099612239 +0295599573917 + 029507069 140 + 0065218663402 + o-62190881458 + 0059161461866 + 0056130696324 + o053098876291 + 0'050066292954 +o 047033237205 + '043999999614 + 0040966870403 +0 037934139418 +0'034902096I05 +0'031871029480 +0'028841228107 + 0'02582980070 + 0022786572947 + 0019762293785 +0 'o6740429070 + 0013721264707 +0'010705085992 + o0007692177584 + 0004682823482 +000o1677306999 - O'OOI324089265 - 0-004321083446 -0'007313394442 - 0010300741939 - 0013282846438 - O'I6259429273 - O019230212645 - 0022194919639 - 0'025153274254 - 0028105001425 -0031049827049 - 0033987478007 - o036917682190 - 0039840168524 - o04275466699I - 0045660908657 - 0oo48558625692 -0'05 1447551397 - 0054327420222 I 256 TABLE I. (continued). x Jo(X) - Jl() 7'20 +0o295070691401 - o'054327420222 7'60 7'21 +0-294513056583 - 0057197967799 7'61 7'22 +0-293926763993 - o-60058930954 7-62 7-23 +0'2933119I0786 -o-o62910047738 7-63 7'24 +0-292668596729 -o0o65751057450 7'64 7'25 +o0291996924192 -o0068581700653 7'65 7'26 +0-29129699813I - 0-071401719205 7'66 7'27 +0-290568926079 - 0074210856276 7-67 7'28 +0'289812818129 -0-077008856374 7-68 7-29 +0-289028786922 -0-079795465364 7'69 7'30 + O288216947635 - 0082570430493 7'70 7'31 +0287377417963 -o0o85333500412 7'7I 7'32 +0-2865I0318III - OO88084425194 7'72 7'33 +0o285615770772 - 0-090822956363 7'73 7'34 +0-28469390III9 - o0093548846906 7'74 7'35 +0'283744836788 -0'096261851305 7'75 7'36 +0 282768707860 - -098961725549 7'76 7'37 +0 281765646852 - 01OI648227162 7'77 7'38 + 0280735788696 - 0'104321115218 7'78 7'39 +0 279679270724 -0-106980150367 7'79 7'40 + 0278596232657 -0-109625094854 7'80 7.41 +0-277486816584 -0-112255712538 7'81 7'42 + 0276351166945 — 0'114871768912 7-82 7'43 +0-275189430519 0-11 7473031128 7-83 7'44 +0'274001756407 - o1 20059268011 7-84 7'45 +0'272788296009 - O'22630250080 7-85 7'46 +0'271549203014 -O'I25185749572 7'86 7'47 +0'270284633379 -0'I27725540456 7'87 7'48 + 0268994745315 - 0130249398456 7-88 7'49 +0-267679699262 - OI32757101068 7-89 7'50 +0'266339657880 -O'I35248427580 7-90 7'51I +0264974786027 -0'137723159089 7'9I 7'52 +0'263585250739 -'0I4018I078522 7'92 7'53 +0'262171221215 -O'I42621970654 7'93 7'54 +0-260732868795 -0-145045622124 7'94 7'55 +0-259270366946 — O'147451821455 7'95 7'56 +0'257783891239 - 0'49840359071 7'96 7'57 + 0256273619329 - 0-152211027316 7'97 7'58 +0-254739730943 -0'154563620468 7'98 7'59 + 0253182407850 -01 56897934760 7'99 7-60 +0'251601833850 -0'159213768396 8-oo Jo(x) +0'251601833850 + 0249998194750 + 0248371678346 + 0'246722474402 +0'245050774627 + 0'243356772660 + o241640664046 + 02399026462I7 + 0238142918467 +o02363668 1936 +0-234559139586 + 0232735496182 + 0230890958266 + 0229025734139 +0o227140033840 + 0225234069120 + 0223308053424 +0o22136220I866 +0'219396731209 + 0217411859839 +0'215407807746 +0-2 13384796501 +0211I343049230 + 0-209282790594 + 0207204246765 + 0205107645402 + 0202993215628 +0'20086 188009 +0O1I987II794526 + 0196545268555 + 0194361844841 + 0o92161759476 +0I189945249872 +0-I87712554741 +O'185463914068 + OI83I99569087 +0'I80919762257 + 0 I78624737238 + 0176314738866 +O0173990013I28 +OI171650807138 -J1(x) - 0159213768396 - o'6I5io92I566 - o163789196464 - o166048397306 -- 168288330341 - 0170508803876 - 0172709628281 - 0I74890616014 - 0177051581630 - 0I179192341800 - 0I81312715325 - 0183412523148 - 0'185491588374 - 01I87549736279 - 0189586794329 - -I91602592189 - 0193596961740 - 0-195569737092 - 0'197520754596 - 0199449852859 - 020I356872756 - 0-203241657440 - 0-205104052360 - 0206943905267 - 0-208761066232 - 0-21055538765I - 0'212326724262 -- 0214014933156 - 0215799873784 - 021750I407969 - 0219179399922 - 0220833716244 - 0-222464225941 - 0'224070800436 - 0225653313572 - 022721164I627 - 0228745663321 - 0230255259825 - 02317403I4769 - '233200714254 - 0234636346854 TABLE I. (continued). 257 x Jo(x) -J1(x) x Jo(x -J, _ i ___ __ ___________ 8-oo S-oi 8'02 8-03 8-04 8-05 8-o6 8-07 8-o8 8-09 8-10 8'I2 8-13 8'14 8'15 8- i6 8-I7 8-I8 8'19 8-20 8-21 8-22 8'23 8-24 8'25 8'26 8-27 8-28 8-29 8-30 8-31 8-32 8'33 8'34 8'35 8'36 8-37 8-38 8'39 8'40 +-0I71650807I38 +o'I69297369III + 'I66929948339 + -064548795169 +o I62154160970 + ' 159746298II7 +0'157325459958 + 0I54891900797 + 0152445875859 +0'149987641274 +0-147517454044 + 0145035572024 + 0I42542253891 +01I40037759122 + '1 37522347965 +0 134996281417 + 0132459821198 +0-I2991322972I +0'I27356770071 +0 124790705977 +O 122215301784 +0- 19630822433 + 0' I 7037533429 + 0 I114435700818 +0'III82559II6I + 0I0920747 506 +0-106581609366 +0-103948272687 +- II01307729828 + 009866024953I + o-09606100895 +0'093345553353 + o0ogo678876643 + -o088006340781 +0-085328216040 + 0O082644772917 + 0'079956282113 +0'077263014501 +0'074565241107 + 0071863233078 + 0069157261657 - 0'234636346854 - 0236047103631 -0'237432878137 - 0238793566425 - 0240129067056 - 024143928110 IOI - 0242724112158 - 0'243983466348 - 0245217252327 - 0246425381291 - 0-247607766982 - 0-248764325692 - 0-249894976273 - 0250999640134 0-252078241253 - 0253130706180 -0-254156964039 -0-255156946534 -0'256130587952 - 0257077825169 - 0257998597649 - 0258892847451 - 0259760519231 - 0260601560243 - 0261415920344 - 0262203551993 - 0262964410256 - 0263698452805 - 0264405639923 - 0-265085934502 - 0265739302042 - 0266365710658 - 0266965131077 - 0267537536636 - 0268082903285 - 0-268601209586 - 0269092436712 - 0269556568447 - 0269993591184 - '270403493925 - 0270786268277 8-40 8-4I 8-42 8-43 8'44 8-45 8-46 8-47 8'48 8-49 8-50 8-51 8'52 8'53 8'54 8'55 8-56 8-57 8-58 8'59 8-60 8-6i 8-62 8-63 8-64 8-65 8-66 8-67 8-68 8-69 8-70 8-71 8-72 8-73 8'74 8-75 8'76 8-77 8-78 8'79 8-8o +0oo69157261657 +00o66447598160 + -063734513946 +0oo06118280395 +0 058299168877 +0'05557745073I + 0-052853397237 +0'050127279588 + 0047399368869 + 0-044669936026 +0'041939251843 +0 039207586917 +0'03647521 I629 + 0033742396123 + 0031009410275 +0'028276523672 +0'025544005583 + 0'022812124938 +0-020081150296 + 017351349826 + 0-014622991279 +0 o- 1896341961 +0'009I71668713 -+0 006449237878 +0- 003729315286 +0 001012166219 - 000 I701944606 -- 004412753067 - 0007119995658 - 0009823409518 -0'012522732450 — 0oI5217702949 - 0-017908060228 - 0020593544236 - 002327389569I - 0-025948856095 - 00286I8167764 - 003128I1573850 - 00339388 I 8366 - 0-036589646207 - 0039233803177 - 0-270786268277 - 0'271141908453 - 0-2714704 11269 - 0'2717I7777614 - 0-272046005084 - 0272293102707 - 0-272513076214 - 0272705935396 - 0-272871692631 -0-273010362878 - 0-273121963674 - 0'2732065 5132 - 0273264039934 - 0'273294563325 - 0'273298i 13 12 - 0273274719657 - 0273224415870 -0-273147237207 - 0-273043221660 - 0272912409756 - 0-272754844546 - 0'27257057 599 - 0272359639000 - 0'272122097337 - 0271857999697 - 0271567401658 - 0'271250361281 - 0270906939104 -0-270537198130 - 0-270141203821 - 0-269719024092 - 0-269270729296 - 0268796392222 - 0-268296088078 - 0-267769894490 - 0'267217891486 - 0-266640161489 - 0-266036789304 - 0'265407862113 - 0-264753469460 - 0'264073703240 17 G. M. 258 TABLE I. (continued). - Jo(x) j -J1 (X) xz Jo() - J, () I.- - I 8'80 8'8I 8'82 8'83 8'84 8'85 8'86 8'87 8-88 8-89 8-90 8'9I 8-92 8'93 8'94 8'95 8'96 8'97 8'98 8'99 900oo 9'0I 9'02 9'03 9'04 9'05 90o6 9'07 9'08I2 9'1I 9'I6 9'12 9'13 9'20 9'20 I - 0039233803177 - o04187 1036007 - 0o44501092388 - '047123720982 - 0'49738671456 - '052345694498 - 0'o54944541843 - 0057534966296 - oo6 I I 672 I 752 - 0'062689563221 - o06525324685 I - o0o67807529947 - 0'070352170997 - o0o72886929689 - 0'075411566939 - o'077925844909 - 0-080429527028 - 0o082922378016 - o0o85404163904 - o0o87874652054 - oo9033361 1183 - o09og27808I 1380 - 0-095216024131 - o09og76390o22336 - 0-100049580330 - 0 I 2447473906 -0 104832480333 - 0107204378374 - o'I095629483 I0 - o0I I I90797I956 - o0 114239232683 - 0- I 116556515436 - 0I I 8859606752 - '02I1148294781 - 0' I 23422369306 - O'I2568162I757 - 0' 127925845233 - 0'130I 548345 I 9 - 0-I32368386I05 - 0-134566298203 - O' I36748370765 0-264073703240 9-20 - O' I 36748370765 - 0-2 I 740865496o 0-26336865769i 9'21 - O' 1389 I4405500 -O'2I5795Oi6778 -0-26263842938i 9-22 -O'I4io64205893 -O'2I4i6i8i6342 -O-26i883II7iq6 9'23 O'I431975772I9 - 0-2125092337o6 0-26i 102822332 9'24 O' I45314326565 - O-2IOS3745o6l2 0-260297648278 9'25 O' I 47414262S4 I - 0-209146650470 -0-2594677oo8O7 9-26 O' 149497 I 968o I - 0-207437Oi8341 - 0-2586 I 3o87962 9-27 O' I 51562941057 - 0-2057o8740917 - O'25773"920049 9-28 - 0-1 536i I3ioo96 - 0-203962oo6501 j - O' 2568303o96 I 5 9-29 - O' I 55642120296 - 0-202 I 97004987 - O'255902371444 9-30 - 0-157655189943 - 0-2004I3927844 -O'254950222539 9-3I - O'I59650339244 -o-i986I2968ogi - 0-2539739821 IO 9'32 - o- I 6 I 627390345 o- I 9679432028 I -0-25297377I56i 9'33' -o-i63586i67343 O' 194958i8O48i - 0-251949714476 9'34 -o-i655264963o6 0-193104746248 - 0-2509O I 936605 9'35 - o- I 67448205283 -0-19I2342i6615 0-249830565850 9-6 - o- i6935I 124322 j - O' 189346792o63 - 0-248735732253 9-37 -O'I71235o8548i - o- I 874426745 07 - O'247 6 I 7 56797 6 9-38 -0-173099922846 - O' I85522o67274 - 0-246476207294 9-39 - 0-174945472543 - o- I 8358517 5079 -0-2453II786573 9-40 -0-17677I572752 o- I 8 I 63 2204007 - 0-244I 2444426 I 9-41 O' I 78578o637i8 O' I 79663361493 - 0-2429I432o868 9'42 o-i8O364787772 O' I 77678856298 -0-24i68I558953 9'43 -o-i82I3I589336 O' I 75678898489 0-2404263031 I 1 9-44 -0-183878-I4938 O' I 73663699419 -O'239148699952 9'45 O' 1856o4813228 - O' I 7 I 63347 I 704 - 0-237848898oSS 9-46 O' 187 3 I 0934989 - o- I 69588429202 - O' 2365 27048 I I 9 9'47 -o-i8899651-)3147 - o-i67528786993 - O' 23 5 1 8 33026 I 2 9'48 -o-igo66I462784 - o, I 6545476135 3 -0-2338178i6o88 9'49 - O'I 92305 58 I I 54 - o- i63366569738 -0-232430745oo6 9'50 - O' I 93928747687 - o- I 6 I 264430758 -0-23IO22247743 9'51 - O' I 95 53o8240 I 0 -O'I59I48564154 - 0-22959248458 I 9'52 - O' I 97 I I I 673948 - O' I 570 I 9190783 — 0'22SI4i6I7686 9-53 -o-i9867li63543 - O' I 54876532586 - O' 2266698 I I 094 9-54 - O' 200209 I 6 I o6o -O'I5272o8I2575 - 0-225 I 7723o692 9-55 - 0-20I725537001 - 0-150552254803 - 0-223664044201 9-56 - 0-20322Oi64I 14 — 0'14837io84348 - O' 222 I 30421 I 59 9-57 - 0-2046929 I 7400 -O'I46I77527286 - 0-220576532901 9-58 - O' 2o6 I 4367412 7 -0-14397i8io670 - 0-219002552542 9-59 - 0-207572313841 -0-141754i625o8 - O' 2 I 74o865496o 9-6o - 0-2o89787 I 8369 - O' I 39524811741 - 0'264073703240 - 0'263368657691 - 0262638429381 - 0'26188311I796 - 0'261102822332 - 0'260297648278 - 025946770o807 - 0'258613087962 - 0'257733920049 - 0'256830309615 - 0255902371444 - 0'254950222539 - 02539739821 Io - 0252973771561 - 0251949714476 - 0'25090I936605 - 0'249830565850 - 0o248735732253 - 02476I7567976 - 0'246476207294 - 02453 I 11786573 - 024412444426I - 0242914320868 - 024I68I558953 - 0'2404263031 I I1 - 0239148699952 - 0o237848898088 - O'236527048 I9 - 0'235 I833026I2 - 02338178I6088 - 0232430745006 - 0231022247743 - 022959248458I — 0'228I4I6I7686 - 0'2266698I I094 - 0'225 I 177230692 - 0'223664044201 - 0o222I304211I 59 - 0220576532901 - 0'219002552542 - 0'217408654960 I 9'20 9'2I 9'22 9'23 9'24 9'25 9-26 9'27 9-28 9'29 9'30 9'3I 9'32 9'33 9'34 9'35 9'36 9'37 9'38 9'39 9'40 9'41 9'42 9'43 9'44 9'45 9'46 9'47 9'48 9'49 9'50 9'5I 9'52 9'53 9'54 9'55 9'56 9'57 9'58 9'59 9'60 I - 0' I 36748370765 - 0 I 38914405500 - '14o1064205893 -- OI431975772I9 - o 145314326565 - 0'I47414262841 - o14949719680 I - o'I51562941057 - o1 536I 13Ioo96 - 0I 55642120296 - 'I57655189943 - OI59650339244 - o'161627390345 - o 163586167343 - 0165526496306 - o0I67448205283 - o-I6935I1124322 - -I71235o8548I - 0'I73099922846 - 0.174945472543 -0'176771572752 — o'178578o63718 - 0I80364787772 - o I82I31589336 - o0'83878314938 - 0'185604813228 -- 0oI873I0934989 - oI88996533147 - o019066I462784 - 0I9230558II54 - 0 I93928747687 - oI1955308240I0 - o I97 I 111673948 - 0o98671163543 - 0'20020916i060 - 0'20I725537001 - 0o203220I641I4 - 0-204692917400 - '206143674127 - 0207572313841 - 0-2089787I8369 - 0-2 I 740865496o -O'2I5795Oi6778 -O'2I4i6iSi6342 - 0-2125092337o6 - O-2IOS3745o6l2 - 0-209146650470 - 0-207437Oi8341 - 0-205708740917 - 0-203962oo6501 - 0-202 I 97004987 - 0-2004I3927844 -o-i986I2968ogi o- I 9679432028 I O' 194958i8O48I O' 19310474624S -0-19I2342i6615 - O' 189346792o63 - 0-187442674507 - O' I85522o67274 - O'IS3585175079 o- I Si 63 2204007 O'I 79663361493 O' I 77678856298 O' I 75678898489 O'I 73663699419 O'I 7 I 63347 I 704 o- I 69588429202 o-i67528786993 o' I 6545476135 3 o, i63366569738 o- I 6 I 264430758 O'I59I48564154 O' I 57OI9190783 O' I 54876532586 -O'I5272o8I2575 0-150552254803 0-148371084348 -O'I46I77527286 -0-14397i8io670 -0-141754i62508 - O' I 39524811741 - 0-2I7408654960 - 0'2I57950I6778 - 0'2I4I6I1816342 - 0'2125092337606 - 0o2I837450612 - 0'209146650470 - 0o207437oI8341 - 0205708740917 - 0'203962006501 - 0202I97004987 - 0'200413927844 - o0I986I2968091 - o I9679432028I -O' 94958I8048I - 0'193I04746248 - 091I2342I6615 - O'I89346792063 - o087442674507 - O I85522067274 - 0'I83585175079 -- O'I8632204007 -0'179663361493 - 0oI77678856298 - OI 75678898489 - O'I73663699419 - 0'I7I63347I704 - 0oI69588429202 -O-i67528786993 - O'I65454761353 -O' 63366569738 - O'I6I264430758 -0'I59I48564154 - O'I57019190783 - O154876532586 - O-152720812575 - 0'150552254803 — O'I48371084348 - 0'146177527286 - O'14397i8IO670 - 0141754162508 - O' I39524811741 I I I I TABLE I. (continued). 259 x Jo(z) x Jo() -J,(x) 9'60 9-6I 9-62 9'63 9'64 9'65 9-66 9'67 9-68 9-69 9'70 9.71 9-72 9'73 9'74 9'75 9'76 9'77 9'78 9'79 9-80 9'81 9-82 9'83 9'84 9-85 9-86 9'87 9-88 9-89 9'90 9-91 9'92 9'93 9'94 9'95 9-96 9'97 9-98 9'99 I0'00 - 0-208978718369 - 0o210362771833 - 0-21724360660 - 0'213063373585 - 0214379701667 - 0215673238291 - 0216943879179 - 0-21819522398 - 0219416068367 - 0220617419863 - 0221795482032 - 0-222950162390 - 0-224081370836 - 0225189019654 - 0-226273023521 - 0'227333299512 - 0-228369767107 - 0-229382348196 - 0'230370967084 -0-231335550495 - 0232276027579 - 0233192329916 - 0'234084391517 -o 234952148834 - 0'235795540759 - 0'236614508629 - 0-237408996230 - 02381 78949800 - 0238924318032 - 0'239645052073 - 0240341105535 -0-241012434487 -0-241658997463 - 0242280755465 - 0242877671958 - 0243449712877 - 0-243996846626 - 0244519044079 - 0-245016278580 - 0245488525942 - '24593576445I - 139524811741 - 0- 37283988215 -0-135031922668 - 0132768846695 - 0'130494992737 - 0-128210594048 - 0125915884679 - 0123611099451 - 0121296473933 - o 118972244417 - - 116638647900 - oI 14295922054 - I 111944305207 - 0-109584036317 - 0107215354950 - 0104838501258 - 0-102453715952 - 0-100061240280 - 0-097661316004 - 0095254185376 - 0-09284009 I 13 - 0090419276375 — 0087991984743 - 0085558460188 - o083118947058 - o-o80673690044 - 0078222934162 - 0075766924729 - 0073305907338 - 0070840127831 - o-o68369832284 - 0o65895266972 -o o63416678354 - oo609343 3045 -o 058448417794 - '055959239457 - 0053467024979 -0o050972021363 - 0048474475654 - 0'045974634906 - 0043472746169 i I0'00 10'01 10'02 IO-03 10-04 10-05 io-o6 10-07 IO'O8 10-09 10*10 IO-II 10'12 10-15 IO'I4 IO'I5 IO-I6 I0197 10'20 10'21 10'22 10'23 10'24 10'25 10'26 10'27 1028 1029 IO030 io'3o 10-31 10'32 10'33 Io'34 Io'35 10-36 Io'37 10-38 10'39 10'40 - 0245935764451 - 0246357974862 - 0246755140400 -0-247127246760 - 0-247474282103 - 0o247796237059 - 0248093104724 - 0-248364880658 - 0248611562881 - 0-248833151876 - 0-249029650581 - 0249201064392 - 0249347401155 - 0249468671 67 - 0249564887171 - 0'249636064351 -o 249682220330 - 0249703375168 - 0249699551355 - 0-249670773804 - 0249617069854 - 0-249538469258 -o 249435004182 - 0249306709197 - 0249153621275 - 0-248975779783 - 0248773226477 - 0248546005495 -0-248294163353 - 0248017748933 - 0o247716813482 - 0-247391410602 - 0247041596243 - 0-246667428695 - 0246268968580 - 0-245846278846 - 0245399424757 - 0'244928473884 - 0'244433496098 - 0243914563561 - 0243371750714 - 0o43472746169 - o0 40969056455 - 0038463812722 - 0035957261846 - o'033449650599 - 0-030941225625 - 0028432233416 - 0-025922920290 -0-023413532364 - 0020904315537 - 'o18395515458 - 015887377509 - 0-I3380146780 - 0oo10874068044 - 0oo8369385737 - 000586634393I - 0003365186314 - 0ooo866156165 + 0001o630503669 + 0004124550795 + ooo6615743298 + 0-009103839761 + o I I588599292 + 0-014069781546 +0o016547146743 + 0-019020455697 + 0021489469834 +0'023953951217 + 0026413662567 + 0-028868367285 +0-031317829476 + 0'033761813968 + 0'036200086339 + 0038632412933 + 0-041058560885 + 0043478298146 +0'045891393496 +0'048297616575 + 0050696737897 +0o053088528877 +0'055472761849 17-2 260 TABLE I. (continued). X | Jo,(X) — J1(x) x JO(x) -J1(x) --- I { -- I I I 10 40 IO'41I 10'42 IO'43 10'44 IO'45 IO'46 Io'1047 IO'48 10'49 IO050 10'52 1053 10'54 1o'55 10'56 IO'57 IO 58 10'59 IO060 io-6I 1062 Io'63 IO'64 Io'65 Io'66 I O67 io-68 Io'69 1070 /O'71 10'72 Io'73 IO'74 10'75 IO 76 10O77 IO078 Io079 io'8o - o024337I750714 - 0242805I34273 - 0-242214793214 - 0'241600808767 - 0o240963264405 - 0-240302245833 - 0o2396I7840978 - 0'238910139979 - 0o238179235 I77 - 0237425221 10I - 0-236648194462 - 0o235848254136 - 0235025501 I 55 - 0-234I80038696 - 0-2333I1972068 - 0-23242140870I - 0'23150845813I - 0'230573231989 - O'229615843992 - 0o228636409922 - 0227635047621 - 0-22661187697I - 0'225567019886 - 0'224500600296 - 0O223412744130 - 0o2223035793 I - 0-22I 173235728 - 0-22002I845238 - O0'21884954I635 - o21I765646o650 - 0.2 I 6442739924 - 02 15208519001 - 0o2 13953939309 - 0.212679144146 - 021I 384278663 - 0-2I00o69489850 - 0o208734926518 - 0-207380739286 - o'206oo7080560 - 0204614104523 - 0o203201967I 12 2761849 IO-80 -0-203 +0'142i66568299 + O'05547 2OI967II2 +0'0578492IO087 IO-SI - 0-20I 77o826005 + O' I 44-05 8996415 +o-o6O2I7647828 IO-82 - O'2OO32OS4o6O3 +O'I45935398812 +o-o62577850293 IO-83 - O'I98852I72OI4 + O' I47795605727 + oo64929593703 IO-84 O' I 9736498 3034 +O'I49639449I22 +o-o6727265530S IO-85 0-1 I9585943813 I +O'I51466762702 * o-o696o6813400 io-86 O' I 94335 703428 +0"532773SI926 * 0-071931847339 IO'87 O' I 92793946683 + O' 15507 I I44022 * 0-074247537568 IO-88 -O'l9I234337275 + O' I 56847888004 * 0-076553665638 io-89 o- i89657046 I 8 I +o-i586O7454682 * 0-078850OI4227 10-90 O' I 88o62245963 +o-i6O34968668i +o-oSii 6-67I58 io-gi -o-i86450II0748 + o- I 62074428448 3 3 +0'08341250942I IO'92 - O' I 848208 I 6208 +o-i6378I526274 +0'085678227I91 IO-93 - o- I 831745 39542 +o-i6547OS28298 +O-087933307849 10-94 - O' I 8 I 51 I 45946 I +o-i67I42I8452S * 0-090177540002 IO-95 -O'I7983I756i65 +o-i68795446850 * 0-092410713500 io-96 — O'I78I356il325 +O-170430469041 +0'0946326I9458 10-97 - O' I 764232O8o66 + O' I 7 2047 I o67S3 + o-o96843050272 IO'9S -O'I74694730946 +0"736452I7675 +0'09904I799642 IO-99 - O' I 72950-65937 +O'I7522466I243 +O'IO12286625S6 II'00 - O' I 7 I I 90300407 +O'I76785298957 + O' I03403435462 I I -oi - o- I 694147 23099 + 0-178326994235 +O'IO5565915987 II-02 -o-i67623824II3 +O'I798496I2465 +0'1077I5903254 II-03 - o- i65817794883 + O' 18I 35302IO05 + O'I 098 5 3 I 97 747 11'04 -o-i63996828i6i + 0-182837089204 + O'I I I 9 7 76o 1 366 II'05 -o-i62i6iII7996 +O'IS43Oi6884o6 +O'II4088917441 I I -o6 o- i6O3IO8597I2 +0'18574669i96-I + o- I I 6 I 86950748 II-07 O' I 5 8446249891 +O'I87I7I97526o +O'II827I50753I II-08 O' I56567486350 +O'I885774I5689 +O'I203423955I5 II-09 -O'I54674768I22 +o-i89962892696 * 0-122399423927 II-IO O' I 5 2768295436 +O'I9I328287775 * O' I 244424035 I 3 II-II O' I 50848269694 +O'I9267,-48448o +O'I2647II46550 IVI2 O' I489I4893455 + 0-193998368432 +O'I2848546687I II-13 O' 1469683 704 IO +O'I95302827334 +O'I3o485I79874 IVI4 0 - I 45 008905 36o +o-i96586750976 +O'I32470IO2543 II-15 O' I43036704202 +O'I9785003I243 +O'I34440053463 ii-i6 O' I 4 I 051973900 +0'199092562I27 +O'I36394852837 IVI7 O' I 39054922470 +O-200314239736 +O'I38334322500 II-IS O' I 37045758956 +O'2OI514962299 + O' I 4025828593 7 II'19 O' I 35024693407 +O-202694630176 +O'I42i66568299 11-20 O' I 3 299 I 93686o +0'203853I45865 +0'055472761849 +o 00578492o0087 + o-o602 17647828 + 0062577850293 + 0064929593703 +o 0o67272655308 +0o-o696068I3400 -+0'071931847339 + 0'074247537568 + 0076553665638 + 0'078850014227 +o-o8 I I36367I58 + 0'083412509421 + 0'085678227 191 + 0'087933307849 + 0O090177540002 + 0'0924107 13500 +0'094632619458 + 0'096843050272 + 0'o09904I799642 + '1O1228662586 + I03403435462 + 0105565915987 + 0'1077I5903254 + O' I09853I97747 + 1119 I I Ig7601366 +o' 14088917441 d- 0116186950748 + 118 I I27I50753I + O' I203423955 I 5 + 0' 22399423927 + ' I 1244424035 I 3 + 012647 I 146550 + o128485466871 +O'130485179874 + 'I32470IO2543 + I34440053463 + I 36394852837 +o'I38334322500 + 0' I40258285937 + O'142166568299 IO*80 Io-SI I0'82 IO'83 IO'84 IO'85 io'86 IO '87 IO'88 IO'89 IO-gO I1'92 IO093 10'94 IO95 IO096 0I'97 Io098 IO'99 II'O0 I I *01 I 1'02 II'o3 II o04 I I'o05 I I *o6 II 0o7 I i*o8 I I 09 ]I'I II'II I I 14 1115 ii i6 1117 II-iS II'I9 1 I'20 I - 0'20320196712 - 0-20I770826005 - 0200320840603 - 0.198852172014 -- I 97364983034 - o I9585943813I - 01I94335703428 - 0'192793946683 - 019I234337275 - 0'18965704618I - 0I 88062245963 - 01I86450110748 - 0'I84820816208 - 0oI83174539542 - 018151 I1459461 - 'I79831756165 — o'I78I35611325 - 01 76423208066 - 0'I 74694730946 - 0'I72950365937 - oI 171 I90300407 - 0.169414723099 - 0.167623824 13 - 01I65817794883 - 0I63996828 6 I - o-62I6I I 17996 - OI6031o8597I2 -- I 58446249891 - 01 56567486350 - 0I54674768I22 - 0 152768295436 - 0I 50848269694 - 0I 489I4893455 - o1469683704 I - I 45008905360 - I143036704202 -- 0'I4I051973900 - I 39054922470 - 0I 37045758956 - o I35024693407 - oI 3299I936860 + 0'42166568299 + I44o058996415 +- 145935398812 + I147795605727 + I49639449I22 + 'I51466762702 + 015327738I926 + O' 15507 I 144022 + I 56847888004 + 158607454682 + oI60349686681 + - 0I 62074428448 + o163781526274 + 0165470828298 +o-167I42I84528 +O- 68795446850 + O-170430469041 + ' I 72047 IO6783 +0o1736452I7675 +- I7522466I243 + I 76785298957 + 0O178326994235 + I798496I2465 + o0'18353021005 + 0 I 82837089204 + O' I84301688406 + 0' 18574669 1967 + 0' I8717I975260 + o I885774I5689 + o189962892696 + 191328287775 + 'I9267348448o +o I93998368432 + o195302827334 + I96586750976 +o'I9785003I243 + 0'99092562127 + '2003 14239736 + 0'20I514962299 + '202694630I76 + 0'203853 145865 I I{ I TABLE I. (continued). 261 x Jo (x) -IJ() Jo () - J, (x) -_ {___ I I'20 I I'21 I I 22 11 23 11 '24 I 'I25 I I '26 II 27 11'28 11 29 I I30 I I '31 I '32 11'33 11'34 11'35 11'36 11'37 11'38 II"39 1 140 11 41 11 42 I I'43 I '44 11-45 11 '46 11'47 11-48 I '49 II-50 11'51 1152 11-53 11-54 11I55 11-56 1157 11'58 1159 - 0132991936860 -01I3094770I3I5 - 0128892199715 - 0- 26825645926 - 0I124748254710 - 0'122660241711 - 0120561823424 - 0 118453217184 -0- 11633464I 133 - 0-114206314208 - O112068456110 - 0109921287289 - 0107765028918 - 0105599902872 - 0 103426131706 - 0-101243938632 - 00o99053547496 - o096855182759 - 0-094649069469 - 0092435433245 - 0-090214500248 - o087986497 63 - 0085751651176 - 0083510189950 - oo8126234160o - 0-079008334679 - 0076748398145 - 0074482761342 - 0-072211653982 - oo69935306115 - 0-067653948112 - 0065367810637 - 0o63077124631 - 0-060782121280 - 0058483032003 - o0o56180088419 - 0053873522332 - 0051563565704 - 0'049250450632 0- oo46934409328 + 0203853145865 + 0204990414012 + 0'206106341416 + 0207200837037 + 0208273812006 + 0209325179625 +0'2I0354855380 +0'211362756947 + 0212348804193 +0'213312919188 +0'214255026208 +0o215175051739 +0'216072924488 +0'216948575381 + 0217801937572 +0'218632946448 +0'219441539632 + 0220227656988 + 0220991240623 + 0221732234896 + 0222450586415 + 0223146244045 +0'223819158911 + 0'224469284397 +0'225096576153 + 0225700992096 +o0226282492413 + 0'226841039560 + 0227376598268 + 0227889135543 +0-228378620665 +0-228845025194 + 0229288322968 +0'229708490101 +0'230105504990 + 0230479348310 +0-230830003018 +0'231157454348 +0-231461689817 +o'231742699216 II'6O I I 6o ii6i II162 11-63 1164 II165 II-66 11'67 I1'68 11'69 11'70 11'71 11'72 II'73 11'74 11'75 11'76 11'77 1178 11'79 II'8o II-8I I1'82 1183 11'84 1185 ir86 1187 II'88 II89 11'91 I I '92 11'93 I '94 11'95 1196 11'97 1198 11'99 - o0-44615674094 - 0-042294477301 - o'039971051364 - 0-037645628720 - 0035318441806 - 0032989723038 - 0o030659704782 - 0028328619340 - 0025996698919 - 0023664175616 - 0021331281388 - 0018998248037 -0I016665307180 - 0014332690232 -0-012000628381 - oo009669352567 - 0-007339093458 - 0'005010081428 - 0'002682546537 - 0000356718505 + o'o0967173307 + 0004288899920 + 0006608232761 +0 008924943683 + 00'11238804987 + '013549589443 +'o015857070317 +0OI18161021385 +0'020461216961 +0'022757431916 +0'025049441700 +0'027337022362 + 0029619950574 + 0031898003653 + 0034170959578 +0'036438597013 + 0038700695332 + 0040957034634 +0'043207395768 +0'045451560353 +0'232000474620 + 0232235010376 +0'232446303109 + 0232634351719 +0'232799157379 + 0232940723529 + 0'233059055883 +0'233154162418 +0'233226053376 + 0233274741260 + 0233300240831 +0'233302569105 + 0233281745349 +0'233237791079 + 0'233 170730054 + 0233080588274 + 0232967393973 +0'232831177619 + 0232671971904 + 02324898 11743 +0'232284734267 + 0232056778820 + 0231805986948 + 0231532402401 +0'231236071121 + 0230917041237 +0'230575363062 + 0230211089083 +0-229824273953 + 0-229414974489 + 0-228983249662 +0'228529160587 + 0228052770520 + 0227554144849 +0'227033351083 + 0-226490458847 +0-225925539874 +0'225338667993 + 0224729919124 +0'224099371266 I 6o - 0'044615674094 + 0-232000474620 12-00 + 0-047689310797 +0-223447IO4491 262 TABLE I. (continued). x Jo(X) - J (z) Jo() - J1() I - -- I II 12'00 I2'01 12'02 12'03 12'04 12'05 12'o6 12'07 12'08 12'09 12'10 + 0-47689310797 + o'049920430320 + 0052144702973 +0-054361913660 +0'056571848157 + 0058774293132 +o-o60969036 67 +00-63155865777 + o065334571427 + 0o67504943560 + oo69666773607 I + 0223447I04491 + 0222773200930 + 0222077744768 + 0221360822234 + 0220622521586 +0'219862933107 + o'29082149091 +0'218280263834 +02 17457373624 +0'216613576726 + 0215748973377 I2'I +0-071819854013 +0-214863665770 12'12 12-13 12'14 12'15 12-16 12'17 I2-I8 12'19 12'20 12'21 12'22 12'23 12'24 12'25 12-26 12'27 12-28 12-29 12-30 12'31 12-32 12'33 12-34 12'35 12'36 12'37 12'38 12-39 +0'073963978255 + 0-076098940860 +0-078224537427 + 0080340564642 +0'082446820302 +0'08454310333I + o-86629213798 + 0088704952938 +0-09077012317I +0-092824528115 +0-094867972612 + 0096900262741 +0-098921205837 +0I1009306105II +0-102928286663 +0'I04914045507 +0-106887699579 +0 108849062765 + I 10797950308 +0 112734178832 + O114657566356 +0-1165679323II + ' 118465097559 + O I20348884405 +0'I22219II6616 +- 0124075619437 + 0'125918219608 +0'127746745377 +02 13957758045 +0'213031356277 +0-212084568463 +0'211117504511 +0'2I0130276228 + 0209122997309 +0'208095783320 + 0207048751691 +0'205982021700 + 0204895714458 + 0203789952902 + 0202664861776 +0-201520567620 + 0200357198756 +0-199174885273 +0-197973759015 +- '196753953565 +0 95515604234 + OI94258848041 +0192983823702 +0-191690671617 +01 90379533851 + 0189050554121 + o187703877780 +O1 8633965I802 + 0184958024768 + 0183559146848 +0'182143169785 12'40 12'41 12'42 12'43 12'44 12'45 12'46 12'47 12'48 12-49 12-50 12'51 12'52 12'53 12-54 12'55 12'56 12'57 12'58 12-59 12-60 I2-6I 12'61 12-62 12-63 12-64 12'65 12.66 12-67 12-68 12-69 12-70 12'71 12'72 12'73 12'74 12'75 12-76 12'77 12-78 12-79 +0-129561026518 + O'31360894344 +0'133146181728 +0134916723111 + 0136672354521 +0-138412913587 + 0140138239554 +0I141848173298 + O143542557339 +0'145221235856 + 0146884054700 + O'48530861410 + 0150161505225 + 0151775837096 + 0153373709704 + 0154954977468 + '156519496560 +0-158067124921 + 0159597722266 + -161111150I04 +0-162607271746 +0 164085952318 + 0165547058774 + 0166990459905 + 0168416026353 +0-169823630622 +0O171213147086 +0I172584452006 + 0173937423535 +01-75271941729 +0O176587888562 + 0177885147930 + 0179163605667 + O'80423149549 +0-181663669309 + -I82885056640 + O'84087205211 +O0185270010670 +- 186433370658 +0-187577184813 + 'I80710246883 + 0179260532985 + 0177794184461 +0-176311359192 +0'174812216550 + 0173296917383 + 0171765624000 +0'170218500152 +0o168655711017 +0o167077423179 +0 165483804615 + -163875024675 + 0162251254066 + 0160612664833 + 0158959430343 +0'157291725265 +0155609725554 + 0153913608430 + 0152203552365 + 0150479737058 +0'148742343422 + 0I46991553564 + '145227550765 +0'143450519461 + 0I41660645228 +01 39858114759 + OI38043115846 +0'13621583736I + '134376469238 + 0132525202454 + 0130662229004 +0-128787741891 + '126901935099 + 0125005003575 + O123097143211 +0O121178550823 + 0II9249424132 +0-117309961743 +0O115360363124 + O113400828590 I 12-40 +0'129561026518 +0-180710246883 I2'80 +0o18870o354781 + -111431559278 TABLE I. (continued). 263 x 12'80 12-81 12-82 12-83 12-84 12'85 12-86 12'87 I2-88 12'89 12'90 12'91 12-92 I2'93 Jo (X) + 0188701354781 +0-189805784222 + 'I90890378823 + 0191955046298 + 0192999696401 +o'I94024240934 + 0195028593748 + 0196012670759 + 0196976389945 +0O197919671360 +01I98842437136 + o1997446I 493 +0'200626120738 + 0-20486893280 - J1 () x J (x) - J, (x) +o III43I559278 + I009452757129 + 0-107464624869 +0105467365986 +0o103461184712 + 0101446286001 + 0099422875508 +oo09739 I I 5957 I + 009535 1345187 + 0093303639994 + 0091248252250 + 0089185390809 +0-087 115265106 +0-085038085131 12'94 + 0202326859628 + 0o8295406I409 12'95 12-96 12'97 12-98 12'99 13'00 + 0'20345952399 +0-203944106324 + 0204721258250 + 0205477347147 + 0-206212314114 +0'206926102377 + 0080863404982 + 0078766327385 + 0-076663040627 +0'074553757168 + 0072438689899 + 0070318052122 13'20 13'21 13'22 13'23 13'24 13'25 13'26 13'27 13'28 13'29 13'30 13.31 13'32 I3'33 13.34 13'35 13'36 13'37 13'38 13'39 13-40 13'4I 13'42 13.43 13 44 I1345 13'46 13'47 13'48 13'49 13-50 13-51 13'52 13'53 I3'54 I3'55 13'56 13'57 13'58 13'59 13 60 +0'216685922259 +0'216945650832 + 2 17183496687 + 0217399452738 +0-217593514066 + o21776567792I +0'217915943717 +0'218044313033 + 0218150789610 + 0218235379352 +0-218298090319 + 02I8338932728 +0'218357918950 +0'218355063505 +0-218330383064 +02I18283896439 +0-218215624587 +0'218125590599 +021I8013819702 +0-217880339252 + 0'2I7725178732 - '2 I 7548369742 + 0217349946004 + 0217129943348 + 0216888399712 +0'216625355135 + 0216340851750 + 0216034933785 + 0215707647547 +0-215359041426 +0-214989165880 + 0214598073436 + 0214185818679 + 02I3752458244 +0'213298050815 + 0212822657111 +0-212326339882 + 0211809163903 +0211I271195961 +0'210712504851 +0'21OI33161369 + 0027066702765 +0-024878857605 + 0-022690195350 + 0020500932874 +0o-o8311286951 + 0016121474234 + o-03931711237 +0o0117422143o8 +0'009553199615 + 0007364883118 +0'005177480555 +00oo2991207414 +0o-00806278917 - 0o001377090000 - 0003558684713 - 0-005738290927 - 0007915694697 - 0oo10090682449 - 0-012263041002 - 0I14432557586 - o-oI6599oI9864 - 0018762215954 - 0-020921934445 - 0'023077964423 - 0025230095486 - 0-027378117768 - 0-029521821957 - 0031660999316 — '033795441703 - 0035924941590 - 0-038049292086 - 0040I6828695 1 - 0042281720622 - 0044389388228 - 0046491085613 - o-o48586609352 -- 0050675756773 - '052758325976 - 0o54834115851 - 0-056902926099 - 0058964557249 I3'01 +0'207618657300 +0-068192057526 13'02 130o3 I3'04 13o05 13-o6 130o7 13-o8 I3o09 I3'10 13'-1 13'12 13'13 13'14 13'15 I3-16 13'17 13I18 13-19 13-20 + 0208289926385 + 0208939859276 + 0209568407762 +0'210175525783 + 0210761169428 +0211I325296943 +0'2I1867868729 +0-212388847348 + 0212888197522 + 0213365886137 + 0213821882244 +0-214256157060 + 0214668683969 +0'215059438525 + 021542839845I + 0215775543638 +0'216100856151 + 0216404320223 + 0216685922259 + 0066060920168 +0oo63924854454 + 0061784075111 + 0059638797173 +0'057489235957 + 0055335607039 + 0053178126239 + 0051017009592 + 0048852473334 + 0046684733877 +0'044514007788 + 0042340511767 + 0040164462629 + 0037986077278 + 0035805572692 + 0'033623165893 + 0031439073935 +0oo29253513878 +0-027066702765 264 TABLE I. (continued). x Jo.W) -J (x) I I3'60 + 021O133161369 -0-058964557249 I3'61 13'62 13'63 13-64 13'65 13-66 13'67 13-68 13'69 13-70 13-71 13'72 13'73 I3'74 I3'75 13'76 13'77 13'78 13'79 13'80 13-81 13'82 13'83 13-84 13'85 13 86 13'87 13 '88 13'89 13-90 13-91 13-92 13'93 13'94 13'95 13 96 13'97 13-98 13'99 14'oo + 0209533238299 +0' 208912810407 + 0208271954434 + 0207610749084 + 0206929275015 + 0206227614833 +0o205505853079 + 0204764076220 + 0204002372641 +0-203220832633 +0'202419548383 +0o201598613965 +o0200758125328 +o 199898180285 +0o199018878503 +o 198120321493 + 0197202612595 +o-196265856970 + -195310161589 +01I94335635216 +0'193342388402 + 0192330533469 + -191300I84501 + o19o251457328 +0 189184469514 +o 188099340348 + -186996190826 +0I185875143642 +- 0184736323171 +0'183579855458 +0' 82405868205 + 0'18124490755 + o180005854081 + 0178780090769 +0'177537335004 +0'176277722558 +- '75001390777 + 0 173708478559 +0'172399126347 +0 17107347611o - 0o06118810678 - o-o63065488629 - o065I04394233 — 0o67135331522 - 0069158105453 -0-071172521923 - 0073178387788 0-075175510884 - 0077163700040 - 0079142765100 - o08 112516941 - 0083072767489 - 0085023329736 - 0086964017760 - 0o88894646742 - 0090815032981 - o092724993914 - o094624348132 - o096512915397 - o098390516658 - 0100256974070 - 0'1021121II008 - 0103955752084 - 0105787723166 - 0I07607851391 - '109415965181 - O111211894262 - 0112995469678 -0 114766523805 - 0116524890369 - 0' I8270404461 - 0120002902550 - 0121722222501 - 0123428203590 - '125120686515 - o126799513414 - '028464527879 - o'I301557497I - 0131752501232 - 0133375154699 II _ 14'oo00 14-0I 14-02 I4-03 14'04 14-05 I4'o6 14-07 14-08 14-09 14 10 I4-IO I4-II 14'11 14'12 14'13 14'14 14'15 14-16 14'17 14'18 14'19 14-20 14'21 14'22 14'23 14'24 14-25 I4'26 14'27 14'28 14'29 14-30 I4'3I 14'32 14'33 14-34 14'35 14'36 14'37 14'38 14'39 14'40 x I - - Jo(X) + 0171073476110 +o-I6973i671331 + o168373856986 +0o167000179537 +0O165610786908 +0'164205828478 + o162785455058 +0 161349818877 + 0159899073571 + 'I58433374159 + o156952877033 +0'155457739939 +0I153948121961 + -I52424183503 +01I50886086277 + 0149333993280 + 0147768068780 + -146188478301 +0-144595388601 + 0142988967659 + oI41369384657 +0O139736809960 +01I38091415099 + 0136433372759 + -134762856750 + 0'I33080042002 + O131385104536 + -I29678221452 + O'I27959570912 + 0126229332114 + oI24487685284 +0 I227348I1649 + 0'120970893423 + -119196113786 + I17410656869 + 0115614707731 +0-113808452342 + 'II1992077563 + 0110165771130 + 0108329721631 + o-16484118490 -J,(x) - o0I33375I54699 - I134983384921 - - 136577042971 - OI3815598I458 - O'39720054543 — '141269117950 - 0-142803028980 - o144321646527 - 0145824831084 - oI47312444762 - 0148784351297 - '0150240416070 - o-I51680506109 - o1I531044901IO — 0'154512238442 - 0155903623164 -'I157278518033 -0-158636798515 - 0O159978341800 - 0I61303026807 - 0162610734200 - 0163901346396 - 0165174747575 -0 166430823692 -0 167669462485 - 0168890553486 - 0I7009398803I - 0171279659270 - 0172447462171 - 0173597293538 - 01I74729052013 - 0175842638087 - 0-176937954108 - 0-178014904291 - O'I79073394724 - oi80o 13333378 - 0181134630112 - 0182137196684 - o I83120946756 - 0o184085795902 - 0185031661615 11 TABLE I. (continued). 265 x Jo (x) -J, (x) x JO ( x) -, (x) I4'40 14'41 14'42 14'43 I4'44 14'45 14'46 14'47 14'48 14'49 14-50 I4'51 14'52 14'53 I4'54 14'55 14-56 14'57 14-58 I4'59 14-60 I4'61 14-62 14-63 14'64 14'65 14'66 I4'67 14'68 14'69 14'70 14'7I 14-72 14'73 14'74 I4'75 1476 14'77 I4'78 14'79 +o' 0o6484I 8490 +o0Io4629I51946 + 0102765013033 + ooo0089 893564 + 0099009986107 +0-097119483970 + 0095220581177 + oo933 13472454 + 0091398353204 + 0089475419488 + 0087544868010 + o085606896092 +oo083661701655 +008 I709483202 + 0079750439794 + 0077784771035 + 0-075812677046 + 0-073834358450 + 007185o006350 + oo69859852307 +0oo67864068323 +o-o65862866820 + oo63856450617 + 0061845022913 + 0059828787267 + 0057807947575 + 0055782708050 +0'053753273205 + 0051719847828 + 0049682636966 + 0047641845902 + 0045597680133 + 0043550345355 + 0041500047438 + o'039446992407 + o037391386420 +0'035333435752 + o'033273346769 +0o0312II3259I3 +0'029147579677 - o'I85o3661615 - OI 859584633I4 - 0186866122350 - 0187754562014 - 0o88623707542 - 0'189473486 119 - 0-190303826889 - 0191 114660960 - 019190592 1406 - 0I92677543276 - 0-193429463596 - 0194161621377 - 0194873957618 - 0-195566415311 - 0196238939443 - 0196891477005 - 0197523976991 - 0198136390405 -0-198728670261 - 0199300771592 - 0199852651447 - 0-200384268898 - 0-200895585039 - 0201386562994 - 0-201857167913 - 0-202307366980 - 0202737129411 - 0'203146426455 - 0203535231400 - 0-203903519571 - 0-204251268330 - 0204578457081 - 0204885067267 - 0-205171082373 - 0-205436487924 - 0-205681271486 - 0-205905422669 - 0-206108933120 - -2o6291796530 - 0-206454008627 14'80 14'81 14'82 14-83 14'84 14'85 14-86 14'87 14-88 14-89 14-90 14-91 I4-92 14 93 14'94 14 95 14-96 14 97 14-98 14.99 I5-00 I5'oI 15 '01 15'02 15'03 15'04 15'05 15 o6 15 07 I5-07 15'o8 150o9 15'10 I5'II 15'I2 15-13 15'14 I5'15 15'16 5'7i6 I5'I, 15'19 + 0027082314586 +0'0250I5737179 + 0022948053986 + 0-020879471508 + -018810196197 + 0-16740434436 +0-014670392520 +0'012600276630 +010530292822 + 0-008460646998 +0oo06391544891 + 0004323192042 + 0002255793783 + 0'01I89555214 - ooo01875318817 - 00oo3938623732 - oo00600055243 - 00oo8059709376 - 0010117082484 - '012172071276 - 0'014224472827 - 0016274084604 - 0'018320704486 - oo020364130779 - 0-022404162240 - 0'024440598094 - 0026473238057 - 0-028501882349 - 0-030526331722 - 0032546387470 - 0034561851456 - 0036572526126 - o038578214533 - oo40578720351 - o042573847897 - o'44563402147 -- 046547188761 - 0-o485250I4094 - 0050496685220 - 0-052462009949 - o206595567180 - 0206716471994 - 0206816724913 - 0206896329814 - 0-206955292607 - 0-206993621235 - 0207011325670 -0'207008417910 - 0o206984911980 - 0-206940823925 -0O206876171810 - 0206790975716 - 0-206685257736 - 0-206559041974 - 0206412354539 - 0206245223541 - 0-206057679091 - 0-205849753289 -0-205621480228 - 0-205372895984 - 0'205104038614 - '2048 14948148 - 0204505666588 - 0204176237900 - 0203826708006 - 0203457124785 - 0-203067538060 - 0'202657999596 - 0-202228563094 - 0-201779284 182 - 0'201310220408 - 0-200821431239 - 0200312978045 - 0'I99784924098 -O 199237334565 - OI'98670276496 - 0-I98083818818 - 0-I9747803233I 0- 196852989694 - 0I196208765420 I I4-8o +0-027082314586 -0206595567180 I5'20 -0-054420796844 -0'95545435866 I 266 TABLE I. (continued). Jo(x) - J1(X) 15-20 15'21 15 22 15'23 I5'24 15-25 15'26 15-27 15'28 I 529 15'30 15-31 15-32 15 33 I5 34 15 35 - 0'054420796844 - o0I95545435866 -0'056372855242 - 0-58317995271 - 0o060256027869 - 0-062186764798 - o194863079227 - 0194161775523 - 0I93441606594 - O'192702656088 - x I5 35 15-36 15 37 15-38 I5 39 J (X) - 0-082890403582 - 0084719235661 - o0o86538408385 - 0088347746952 - 0090147077648 -J1(x) - 01833603220I7 - O- 182403 I 162448 - 0 81I428468883 - 0180436349242 -- 0I79426913096 - 0.06411oo18670 -0 I91I945009455 - o066025602957 - o067933332015 - o-o69833021097 - 007 724486374 - oI91168753932 - 0190373978539 - 0189560774066 -0I188729233063 15'40 -0-091936227862 — 0I78400271655 I5'4I 15'42 15 43 i5 44 - o0o937 5026Io6 - 0-095483302024 - 0097240886416 -0-098987611250 - 0-77356537757 - 0176295825856 - 0I75218252010 - 0-74123933866 -0-073607544951 -oI187879449832 I15-45 -0I100723309676 -0-I730I2990652 -- 0'075482014884 - 0-077347715198 - 0-079204465905 - 0081052088022 - 0187011520415 - o0I8612554258I - o-85221615823 - 0o18429984I336 - 15-46 15 47 15-48 15 49 15'50 - -I 024478I6048 - oIo4160965933 - 0 105862596129 - 0107552544683 -o17188554316o - 0170741713736 - -169581626266 - o-168405406163 - 0082890403582 - 0183360322017 - 0109230650900 -0167213180352 TABLE II. n J,(I) 0 +076519 76865 57966 551 I +044005 05857 44933 516 2 +0'11490 34849 31900 480 3 +001956 33539 82668 406 4 +000247 66389 64109 955 5 +000024 97577 30211 234 6 +000002 09383 38002 389 7 15023 25817 437 8 00942 23441 726 9 00052 49250 i80 10 +o 00002 63061 512 II 11980 067 12 00499 972 13 00019 256 14 689 15 023 I6 00I n Ji(2) 0 +0o22389 07791 4I235 668 I +0-57672 48077 56873 387 2 +0-35283 40286 I5637 719 3 +012894 32494 74402 051 4 +003399 57198 07568 434 5 +o00703 96297 5587I 685 6 +0-0020 24289 71789 993 7 +0 OOo7 49440 74868 274 8 +000002 21795 52287 926 9 24923 43435 I33 I0 +0o 02515 38628 272 II 00230 42847 584 12 00OI9 32695 I49 13 00001 49494 20I I4 10729 463 15 00718 302 I6 00045 o60 17 00002 659 18 148 19 oo8 TABLE II. (continued). 267 lt J. (3) it J. (4) 0 I 2 3 4 5 6 7 8 9 IO II 12 13 14 15 i6 17 18 19 20 21 22 -0-26005 19549 01933 438 +0-33905 89585 25936 459 +o048609 12605 8589I 077 +0o30906 27222 55251 644 +0-13203 41839 24612 210 +0-04302 84348 77047 584 +00-1139 39323 32213 069 +0-00254 72944 51804 694 +oo00049 34417 76208 835 +o-oooo8 43950 21309 092 +0o00001 29283 51645 716 17939 89662 347 02275 72544 832 00265 90696 309 00028 80156 513 00002 90764 476 27488 250 02443 521 00204 983 00016 280 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 -0-39714 98098 63847 372 -0o-6604 33280 23549 I36 +0'36412 81458 52072 804 +0'430I7 14738 75621 940 +0-28112 90649 61360 io6 +0'13208 66560 47098 272 — 004908 7575I 56385 574 +001517 60694 22058 451 +0'00402 86678 20819 004 +0-00093 86oi8 61217 564 +O'OOOI9 50405 5466o 035 +0-00003 66009 12082 608 62644 61794 312 09858 58683 265 01436 19646 909 00194 78845 096 00024 7169I 311 00002 94685 392 33134 523 03525 313 +o0 0000I 228 o88 0o6 +o0 00355 951 00034 I99 )0003 134 275 023 002 268 TABLE II. (continued). n Jn(5) n J (6)... 0 I 2 3 4 5 6 7 8 9 I0 II 12 I3 I4 15 16 I7 18 19 20 21 22 23 24 25 26 27 -0'I7759 67713 I4338 304 -0'32757 91375 91465 222 +0-04656 51162 77752 2I6 +0-36483 12306 13666 994 +0-39123 26304 58648 178 +0'26114 05461 20170 090 +0O13104 87317 81692 002 +0o05337 64101 55890 715 +0'01840 52166 54802 ooi +0-00552 02831 39475 688 +0'00146 78026 47310 474 +000ooo35 09274 49766 209 +00000oooo7 62781 31660 846 +00ooo00 52075 82205 849 28012 95809 572 04796 74327 752 00767 50156 939 00115 26676 659 oooi6 31244 339 00002 18282 584 0 I 2 3 4 5 6 7 8 9 IO II 12 13 14 15 I6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 +-'15064 52572 50996 932 -0-27668 38581 27565 6o8 -0-24287 32099 60185 468 +o11476 83848 20775 296 +0-35764 15947 80960 764 +0-36208 70748 87172 389 +0-24583 68633 64326 55I +0'I2958 66518 41480 713 +0o05653 19909 32461 779 +0'021I6 53239 78417 365 +o-oo696 39810 02790 316 +0o00204 79460 30883 689 +0'00054 51544 43783 2II +0-00013 26717 44249 154 +0-00002 97564 47963 12I 61916 79578 746 12019 49930 6io 02187 20051 176 00374 63692 719 ooo60 62105 141 +0o 27703 301 03343 820 00384 787 00042 309 00004 454 450 044 004 +0o +o 0 00009 29639 841 ooool 35493 798 i88i6 747 02496 677 00316 779 00038 554 00004 415 507 055 oo6 ooI TABLE II. (continued). 269 n J (7) n [J (8) 0 2 3 4 5 6 7 8 9 10 I I 12 I3 14 15 16 17 I8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 +0-30007 92705 I9555 597 -0-00468 28234 82345 833 -0'30141 72200 85940 120 - o'16755 55879 95334 236 +0'15779 81446 61367 918 +0-34789 63247 51183 285 +0'33919 66049 83179 632 +0'23358 35695 05696 084 +0'12797 05340 28212 537 +0-05892 05082 73075 428 +0o02353 93443 88267 135 +000oo833 47614 07687 815 +0-00265 56200 35894 568 +0o00077 02215 72522 133 +0-00020 52029 47759 069 +0-00005 05902 I8514 143 + oooooI 16122 74444 403 24944 64660 269 05036 96762 619 00959 75833 201 + o' 0173 14903 330 00029 66471 543 00004 83925 930 75348 588 11221 932 oi6oi 804 00219 522 00028 933 00003 673 450 +o0 053 oo6 001 0 I 2 3 4 5 6 7 8 9 IO II 12 I3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 +o'I7165 08071 37553 906 +0-23463 63468 53914 624 -0-11299 17204 24075 250 -0o29113 22070 65952 249 -0O10535 74348 75388 937 +O18577 47721 90563 312 +0'33757 59001 13593 077 +0-32058 90779 79826 304 +0-22345 49863 51102 954 +0'12632 08947 22379 605 -+-006076 70267 74251 156 +0'02559 66722 13248 286 +o0oo962 38218 12181 630 +0'00327 47932 23296 605 +o'ooI00 92561 63532 336 +0-00029 26033 49066 572 +0o00007 80063 95467 308 + oooooI 94222 32802 66I 45380 93944 002 09991 89945 347 +0o +0o 02080 58296 397 00411 01536 639 00077 24770 956 00013 84703 619 00002 37274 853 38945 500 06134 520 00928 879 00135 416 001I9 034 00002 583 339 043 005 001 270 TABLE II. (continued). it Jn (9) ')I J. (IO) 0 I 2 3 4 5 6 7 8 9 IO II 12 13 14 I5 16 I7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 -o009033 361 82876 134 +0'24531 17865 73325 272 +0'14484 73415 32503 973 -o-I8093 51903 36656 840 -0-26547 08017 56941 866 -0-05503 88556 69513 708 +0'20431 65176 79704 413 +0-32746 08792 42452 925 +0-30506 70722 53000 137 +0'21488 05825 40658 430 +0'12469 40928 28316 722 +0o06221 74015 22267 619 +00o2739 28886 70559 68i +o-o1o83 03015 99224 863 +0-00389 46492 82756 591 +OO0128 63850 58240 087 +000ooo39 33009 11377 031 +o0OOOII 20181 82211 578 +0-00002 98788 88088 932 74973 70144 148 o I 2 3 4 5 6 7 8 9 IO II 12 I3 I4 I5 16 17 I8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 -0-24593 57644 51348 335 +0-04347 27461 68861 437 +0-25463 03136 85120 623 +0-05837 93793 05186 812 -0-21960 26861 02008 535 -0-23406 15281 86793 640 -0-01445 88420 84785 I05 +0'21671 09176 85051 514 +0-31785 41268 43857 225 +0'29185 56852 65120 046 +0-20748 6io66 33358 858 +0-I2311 65280 01597 669 +0-06337 02549 70156 015 +0-02897 20839 26776 767 +0-01I95 71632 39463 579 +0'00450 7973I 43721 253 +0-00156 6756I 91700 i8i +0-00050 56466 69719 325 +0I00015 24424 85345 524 +0-00004 31462 77524 563 +0oo0000 15133 69247 813 29071 99466 691 06968 68512 289 01590 21987 380 00346 32629 66i +0o 17766 74741 915 03989 62042 141 oo851 48121 408 00173 17662 520 00033 64375 918 oooo00006 25675 712 ooooI 1160o 257 19125 771 03154 368 00501 407 00076 922 ooOII 403 oooo0000 636 227 031 004 +o0 +o0 00072 14634 990 00014 40545 292 00002 76200 527 50937 552 09049 767 01551 096 00256 809 00041 123 oooo6 376 958 140 020 003 TABLE II. (continued). 271 I t J2, (II) 1 I 0 I 2 3 4 5 6 7 8 9 10 I 12 13 14 '5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 -0'17119 03004 o7196 o88 -0'17678 52989 56721 501 +o'13904 75187 78701 270 +0'22734 80330 58067 417 -0-01503 95007 47028 133 -0-23828 58517 83178 787 -0-20I58 40008 74043 49I +o01837 60326 47858 615 +0-22497 16787 89499 910 +0-30885 55001 36868 527 +0-28042 82305 25375 862 +0-20101 40099 09269 403 +0I12159 97892 93162 945 +o-o6429 46212 75813 386 +0'03036 93155 40577 785 +o001300 90910 09293 703 +ooo00511 00235 75677 768 +o'ooI85 64321 19950 713 +ooo0062 80393 40533 526 +0000o19 89693 58159 009 +0-00005 93093 51288 506 +o-ooooI 670Io 10162 830 44581 42060 481 11315 58079 093 02738 28088 453 00633 28125 065 00140 27025 479 00029 81449 927 oooo6 09183 254 ooooI 19846 638 0 I 2 3 4 5 6 7 8 9 I0 II 12 13 14 I5 16 I7 i8 I9 20 21 22 23 24 25 26 27 28 29 30 3I 32 33 34 35 36 37 38 39 40 4I. i n Jn(I2) +0-04768 93107 96833 537 -0 22344 7I044 90627 612 -0o08493 04948 78604 805 +o019513 69395 31092 677 +0'18249 89646 44151 I44 -0-07347 09631 01658 581 -0 24372 47672 28866 628 -0'17025 38041 27208 047 +00o4509 53290 80457 240 +0-23038 o9095 67817 70I +0-30047 60352 71269 311 +0-2704I 24825 50964 484 +0-19528 01827 38832 243 +0'I2014 78829 26700 003 +0o-6504 02302 69017 762 +0-03161 26543 67674 776 +0-01399 14056 50169 178 +0-00569 77606 99443 032 +0o002I5 22496 64919 412 +0-00075 89882 95315 204 +0-00025 12132 70245 400 +00000ooo7 83892 72169 462 +0-00002 31491 82347 716 64910 63105 497 17332 26223 355 04418 41787 923 I0077 81226 324 00252 IOI92 815 00056 64641 343 00012 24800 120 - +o0 22735 384 04164 546 00737 509 00126 418 00020 997 00003 383 529 o8o 012 002 I +0o 00002 55225 904 51329 401 09976 003 01875 946 0034I 699 ooo60 351 ooo0010 346 ooooI 723 279 044 +o0 007 00I 272 TABLE II. (continued). n Jn (I3) nl Jn (14) o I 2 3 4 5 6 7 8 9 IO II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 +0'20692 6I023 77067 811 -0'07031 80521 21778 371 -0-21774 42642 41956 791 +0-00331 98169 70407 051 +0'21927 64874 59067 738 +0'13161 95599 27480 788 -0'11803 06721 30236 362 -0-24057 09495 86i6o 507 -o'14104 57351 16398 030 +0oo6697 61986 73670 624 +0o23378 20102 03018 894 +0-29268 84324 07896 905 +0'26153 68754 I0345 099 +0'19014 88760 41970 970 +-'11876 o8766 73596 841 +0o06564 37814 08852 996 +0'03272 47727 31448 533 +0'01490 95053 14712 625 +0ooo626 93180 91646 024 +0-00245 16832 46768 672 +0o00089 71406 29677 786 + '00030 87494 59932 207 +0oI00010 3576 25487 806 +0-00003 09225 03257 290 90604 62961 o66 25315 13829 722 06761 28691 713 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 +01'7107 3476I 10458 659 +0'13337 51546 98793 253 -O'I5201 98825 82059 623 -o'17680 94068 65096 003 +0-07624 44224 97018 479 +0-22037 76482 91963 705 +o-o8II6 81834 25812 739 - '15080 49196 41267 072 -0-23197 31030 67079 8Io -0'11430 71981 49681 283 +0oo8500 67054 46061 oi8 +0-23574 53487 86911 308 +0-28545 02712 19085 324 +0'25359 79733 02949 247 +o0I8551 73934 86391 849 +o'11743 68136 69834 451 +o-o6613 29215 20396 260 +0-03372 41498 05357 OOI +001I576 85851 49756 457 +0o00682 36405 79731 0o3 +0o00275 27249 95227 770 +0-00104 12879 78062 597 +0-00037 11389 38960 020 +0'00012 51486 87240 324 +0-00004 00638 90543 902 +0-00001 22132 23195 912 35547 63727 213 o9901 84933 738 02645 21017 203 00678 99135 075 +o0 01730 00937 128 00424 90585 590 00100 35431 567 00022 82878 324 00005 00929 928 ooooI 06172 104 21763 505 04319 539 00831 oo8 00155 121 00028 122 00004 956 850 142 023 004 001 +o0 00167 75399 538 00039 95434 356 00009 i8666 897 00002 04185 745 43923 044 09154 753 01850 722 00363 244 00069 281 00012 851 00002 320 408 070 012 002 +o0 +o0 TABLE II. (continued). 273. nf1 J( I5).. _~ - - it J(i6) _- 0 1 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 — 0'01422 44728 26780 773 +0'20510 40386 13522 761 +0'04157 16779 75250 475 -0'1940I 82578 20122 635 -0'II917 89811 03299 529 +0-13045 61345 65029 553 +0'20614 97374 79985 897 +0'03446 36554 18959 165 -01'7398 36590 88957 343 -0-22004 62251 13846 998 -0-09007 I8I1I 47659 054 +0 '09995 04770 50301 592 +0-23666 58440 54768 056 +0-2787I 48734 37327 297 4 024643 99365 69932 593 +0'I8130 63414 93213 542 +o'11617 27464 16494 492 +o-o6652 88508 61974 707 +0-03462 59822 03981 51i +O01657 35064 27580 920 +0-00736 02340 79223 485 +0-00305 37844 50348 374 +0-OOII9 03623 81751 963 +0-00043 79452 02790 717 +0-00015 26695 73472 902 +0-00005 05974 32322 570 +0-0000I 59885 34268 998 0 I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 I6 17 I8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 -0'17489 90739 +0-09039 71756 + 0-8619 87209 -0-04384 74954 -0-20264 15317 -0O05747 32704 +0-I6672 07377 +0-18251 38237 -0-00702 11419 - 0oI8953 49656 83629 185 61304 i86 41292 208 25981 134 26035 I33 37036 433 02887 363 14201 955 52960 653 67162 607 -0'20620 56944 22597 281 -0'06822 21523 61083 994 +011240 02349 26106 790 +0-23682 25047 50244 178 +0'27243 63352 93040 000 +0-23994 10820 12575 821 +0'17745 31934 80539 665 +0-11496 53049 48503 509 + -06684 80795 35030 292 +0-03544 28740 05314 648 +0-01732 87462 27591 996 +0-00787 89915 63665 343 +0'00335 36066 27029 529 +O'00134 34266 60665 86i + 000050 87450 22384 822 +O'ooo18 28084 06488 605 + o-0oo6 25312 47892 069 +0-00002 04181 49160 619 63800 05525 020 19118 70176 952 48294 86476 623 13976 17046 846 03882 83831 6oi +0' +0' - o. 01037 47102 OII 00267 04576 442 o0066 31813 951 00015 91163 08i 00003 69303 606 83013 267 18091 639 03826 599 00786 251 00157 074 00030 535 00005 781 oooo0 067 192 034 oo6 001 +0o +0 - 05505 23866 431 01525 49322 163 00407 7913I 952 00105 22205 645 00026 24966 335 oooo6 33901 280 00001 48351 763 33681 654 07425 886 01591 305 00331 726 00067 325 00013 313 00002 567 483 089 oi6 003 G. M. 18 274 TABLE ] n J (I7) o -0'16985 4252I 5I83 548 I -oo09766 84927 57780 650 2 +o015836 384I2 38503 47I 3 +'013493 05730 49193 232 4 -0'11074 12860 44670 566 5 -018704 4II94 23155 851 6 +000071 53334 42814 183 7 +o18754 90606 76907 039 8 +015373 68341 73462 202 9 -0o04285 55696 90119 084 IO -o019911 33197 27705 938 I -0'19139 53946 95417 314 12 -004857 48381 13422 350 13 +'I12281 91526 52938 702 14 +0'23641 58951 12034 482 15 +026657 17334 13941 622 16 +023400 48109 12568 380 17 +0-17390 79o16 56775 329 8 +0o11381 10104 00982 277 19 + oo6710 36407 80598 906 20 +0-03618 53631 08591 747 21 +00'1803 83900 63146 381 22 +o00838 00711 65064 o 8 23 +0-00365 12058 93489 902 24 +0-OOI49 96624 29085 127 25 +o-00058 31350 82750 457 26 +0-0002I 54407 55475 041 27 +0o00007 58601 69290 845 28 +0'00002 55268 41095 878 29 82282 48436 754 II. (continued). i4 43 44 45 46 47 48 49 J, (I7) + 0o00000 00000 0028 646 00005 752 00001 127 216 040 007 001 b JU (I8) 0 I 2 3 4 5 6 7 8 9 10 II 12 I3 14 I5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 -0-01335 58057 21984 III -0-18799 48854 88069 594 -0-00753 25148 87801 400 +0-18632 09932 90780 394 +0o-6963 95126 51394 864 -0'15537 00987 79049 343 -0'15595 62341 953II i66 +0'05139 92759 82175 233 +0'19593 34488 48114 125 +0-12276 37896 60592 878 -0-07316 96591 87521 246 -0o20406 34109 80060 930 -01'7624 11764 54775 446 -0o03092 48242 92972 998 +0'13157 I9858 09370 005 +0'23559 23577 74215 227 +0'26108 19438 14322 041 +0'22855 33201 17912 845 +0-17062 98830 75068 889 +0'11270 64460 32224 933 +0'06730 59474 37405 969 +0-03686 23260 50899 443 +o001870 61466 81359 398 +o-oo886 38102 81312 419 +0-00394 58129 26439 oo6 +0oo0165 83575 22524 930 +0o0oo66 07357 47241 354 +0-00025 04346 36172 316 +000009o 05681 61275 594 +0-00003 13329 76685 o88 30 31 32 33 34 35 36 37 38 39 40 41 42 +o0 +o 0 25460 o6511 871 07576 56899 262 02172 12767 790 oo6oo 85285 358 ooi6o 59516 541 00041 52780 805 ooo0o 40169 125 00002 52641 372 59563 905 13644 321 03039 452 00658 981 ooI39 163 TABLE II. (continued). 275 n J,, (i8) lz J,, (I9) 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 +0o00ooo 03936 52487 466 33125 31606 465 ioi6i 78601 469 03005 47865 425 oo858 30238 423 00236 99701 950 00063 35269 i6o 0o016 41374 689 00004 12604 562 oo001 00733 463 +0o 23907 III 05520 363 01241 210 00271 949 00058 I04 00012 114 00002 466 490 095 oi8 +0o 003 I0 11I 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3I 32 33 34 35 36 37 38 39 40 4I 42 43 44 45 46 47 48 49 50 51 52 +0-09155 33316 22639 788 -0-09837 24006 77574 175 -0-20545 82166 17725 675 -o'16115 37676 81658 257 -0'01506 79917 88754 044 +0-13894 83060 98231 244 +0-23446 00540 49119 i66 +0o25593 17849 31864 194 +0-22352 31400 39479 918 +o-16758 57435 63992 493 +0o11164 83470 88505 067 +0-o6746 34082 01281 333 +0-03748 12920 93274 722 +0-01933 53734 88407 496 +0-00933 06647 73396 058 +000oo423 68322 54908 86I +o-ooi8i 88937 92153 576 +000oo74 11928 60458 822 +0-00028 76543 37571 496 +o oooIo 66304 50278 219 +0-00003 78491 42225 174 +0-00001 28931 56748 644 42232 64007 245 13325 74644 i8i 04056 79493 594 01193 30911 839 00339 60707 917 ooo0093 62297 I1o 00025 02975 563 oooo6 49605 144 +0 - n Jo (i9) 0 2 3 4 5 6 7 8 9 +0'14662 94396 59651 204 -0-10570 14311 42409 267 -0-15775 59060 95694 285 ooooi 63824 40182 09593 02231 00505 001II 00024 00005 00001 500 226 529 272 913 905 163 o96 051 212 042 oo8 002 +0-07248 96614 +0o18064 73781 +0'00357 23925 -o'17876 71715 - oII647 79745 +0o09294 12955 +01'9474 43287 38052 575 28763 519 10900 486 44079 053 38739 888 68165 452 01405 531 +0 - 18-2 276 TABLE II. (continued). il 0 2 3 4 5 6 7 8 9 IO II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4I 42 43 44 45 I J. (20) +0-I6702 46643 40583 155 +0-o6683 31241 75850 046 -0'16034 13519 22998 150 -o0og890 13945 60449 676 +0-13067 09335 54863 247 +0'-5116 97679 82394 975 -0-05508 60495 63665 760 -0-I8422 I3977 20594 431 -0o-7386 89288 40750 341 +0'12512 62546 47994 158 +0-18648 25580 23945 083 +0-06135 63033 75950 926 -0-II899 06243 10399 065 -0-20414 50525 48429 804 -0'14639 79440 02559 680 -o-ooo8I 20690 55153 748 +0'14517 98404 19829 058 +0'23309 98137 26880 240 +0'25108 98429 15867 351 +0'2I886 J9035 21680 991 +0'16474 77737 75326 532 +0-11063 36440 28972 073 +0'06758 28786 85514 822 +0'03804 86890 79160 535 +0-01992 91061 96554 408 +0oo00978 11657 92570 045 +0-00452 38082 84870 704 +o'oo198 07357 48093 786 +0-00082 41782 34982 517 +0-00032 69633 09857 262 +0'00012 40153 63603 543 +0-00004 50827 80953 368 +0t00001 57412 57351 896 52892 42572 701 17132 43138 017 05357 84096 556 01620 01199 928 00474 20223 i86 00134 53625 859 00037 03555 077 +o 00009 90238 941 00002 57400 689 65Io3 882 16035 615 03849 264 00901 I45 46 47 48 49 50 51 52 53 54 J, (20) +0-00000 00000 00205 887 00045 937 oooI0 015 00002 135 + o0 445 091 oi8 004 001 it 0 2 3 4 5 6 7 8 9 IO0 II 12 13 14 15 16 17 I8 19 20 21 22 23 24 25 26 27 28 29 J, (2I) +0-03657 90710 00862 743 +0-17112 02727 63900 I04 -0-02028 19021 66205 590 -0I17498 34922 24129 740 -0-0297I 33813 26402 907 +0-16366 41088 61690 537 +O-1I764 86712 60541 258 -0-10215 05824 27095 533 -0'I7574 90595 45271 613 -0-03175 34629 40730 458 +01I4853 18055 96074 078 +0-I732I 23254 13181 961 +0'03292 87257 89164 167 -0'13557 94959 39851 484 -0-20078 90540 95646 957 -0-13213 92428 54344 458 +0'OI20I 87071 60869 159 +01I5045 34632 89954 606 +0-23157 26143 56200 203 +0-24652 81613 20674 313 +0-21452 59632 71686 649 +o-16209 27211 01585 97I +0-10965 94789 31485 294 +o-o6766 99966 59621 310 +0-03857 00375 6ioi8 529 +0-02049 00891 94135 328 +0-OI02I 58890 91684 632 +0-00480 63980 80512 332 +0-00214 34202 58204 222 +0oo00090 93892 74698 928 I TABLE II. (continued). 277 iu Jg (21) U J,(22) I 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 +o000036 82263 IOOII 863 +o'ooo14 26858 96763 539 +0-00005 30368 13766 205 +o0ooooi 89501 07095 370 65206 65676 388 21644 29380 552 06940 98925 453 02153 38363 859 00647 12451 956 oo088 59081 313 +0o 00053 00014 oooi4 00003 0000I 35564 351 66878 120 92245 451 02Io3 682 25893 439 06402 159 01544 385 00363 716 00083 679 o00o8 8I8 00004 139 891 I88 039 oo8 002 +o0 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 +0'15657 87523 63312 897 +o-oo668 77613 94996 661 -0'14867 50343 51044 115 -0o19591 05323 87234 626 -0-11847 56916 31548 557 +0-02358 22536 50436 726 +0'15492 09927 27678 042 +0-22992 48253 58490 979 +0-24222 18874 36988 I95 +0-21047 86063 45123 920 +0'15960 09064 94612 017 +o-I0872 32066 44100 I14 +0'06772 94346 70324 584 +oo03905 01053 63880 797 +0-02102 08047 93040 864 +0-01063 54332 37852 155 +00oo508 43495 I8050 788 +0'00230 65473 53549 852 +00ooo99 65480 50398 821 +0-00041 13109 65719 66o +o-oooi6 26010 34811 129 +o'oooo6 17102 26458 170 +0-00002 25296 44563 381 79268 56737 734 26921 72329 410 08838 89067 607 02809 09079 813 oo865 24117 203 0 1 I 2 3 4 5 6 7 8 9 I0 II +0' +0' -o'I2065 I4757 +0'11717 77896 +o013130 40020 -oo09330 43347 -0-15675 06387 +0-03630 41024 +0-17325 25035 +0-05819 72631 -0-13621 78815 - 0-I5726 48133 04867 18o 43851 701 36126 426 28192 351 80178 885 44490 938 27674 766 16058 934 44728 17I 30406 695 00258 58244 815 00075 05863 943 00021 18157 153 00005 81645 I87 ooooI 55546 760 40541 854 10306 278 02557 128 00619 634 00146 728 00033 973 00007 696 ooooi 706 370 079 oi6 003 001 +0-00754 66706 38031 784 +o'I6412 54230 01344 68I 278 TABLE II. (contimued). n1Jn (23) Jn,(23) 0 I 2 3 4 5 6 7 8 9 10 Io 12 13 14 15 16 17 18 I9 20 21 22 23 24 25 26 27 28 29 -o'-6241 27813 13486 542 -0-03951 93218 83701 511 +-015897 63185 40990 759 +0-06716 73772 82134 687 -0'14145 43940 32607 797 -0-11636 89056 41302 6i6 +0oo9085 92176 66824 051 +o'16377 37148 58776 034 + ooo882 91305 08083 IOO -01o5763 17110 27066 051 -0o13219 30782 +0o-4268 12081 + 017301 85817 +0-13785 99205 -0-01717 69323 - 0'15877 09687 - 018991 56355 -0'10545 94806 +0-0340I 90118 d- 015870 66297 68395 662 84982 867 49683 622 97295 695 78827 619 10651 057 04630 281 87095 422 80228 354 17018 062 65279 749 31294 545 74475 507 89441 207 04406 908 02364 513 30 3I 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5I 52 53 54 55 56 57 58 +0-00246 97721 07261 064 + 0000oo 8 54000 46200 88i +0-00045 60888 86845 66o +o000018 37168 56326 172 +0-00007 10986 13916 400 +0-0002 64877 41339 705 95162 51030 530 33022 6i886 301 r1084 17647 134 03603 35556 401 +0o 01135 00347 00103 00029 oooo8 00002 00002 89891 967 59720 oo6 36066 315 89391 752 41659 366 30870 169 61745 644 I6II2 408 04105 067 OI021 783 00248 619 00059 i68 00013 780 00003 142 702 154 034 007 001 +0'22819 + 023814 +0o20668 -o1 5725 +0-10782 + 0o6776 19415 89208 86964 55419 23875 50928 +o0 +0-03949 30316 31168 121 +0-02152 35004 50711 239 +0'01104 04042 09632 179 +0'00535 74837 11871 458 TABLE II. (contibnued). 279 n o I 2 3 4 5 6 7 8 9 10 II 12 13 I4 I5 16 17 I8 19 20 21 22 23 24 25 26 27 28 29 __ -0o05623 02741 66859 267 -0'15403 80651 83121 221 +00o4339 37687 34932 499 +o-I6127 03599 72276 638 -0-00307 61787 41863 339 -0I16229 57528 86231 084 -0o-6454 70516 27399 613 +0-13002 22270 72531 278 +014039 33507 53042 858 -0-03642 66599 03836 039 -0'16771 33456 80919 887 - 010333 44614 96930 534 +0o07299 00893 08733 565 +0-17632 45508 05664 o98 +O'11802 81740 64069 208 -0-03862 50143 97583 355 -0-'6630 94420 61048 403 -0'18312 09083 50481 i8i -0o-931I 18447 68799 938 +0-04345 31411 97281 275 +O'I6191I 26516 64495 289 +0'22640 12782 43544 208 +0-23428 95852 61707 074 +0-20312 96280 69585 428 +O'15504 22018 71664 996 +o-o1695 47756 73744 565 +oo06778 02474 48636 i8o +0'03990 24271 31633 826 +0-02200 02135 97539 927 +-O'1143 14045 95959 338 30 3I 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5I 52 53 54 55 56 57 58 59 60 +0-00562 56808 42695 140 +0o00263 27975 10778 513 +0o-OII7 57127 26816 017 +0'00050 24364 27397 533 +0-00020 59874 48527 199 +0-oooo8 11946 76762 864 +0o00003 08303 58697 822 +0-00001 12963 99330 6oi 40002 05904 866 13709 19368 140 +o0 +o0 +o0 04552 82041 591 01466 87437 162 00459 00035 379 00139 62686 664 00041 32925 I68 00O11 91372 284 00003 34720 896 91724 484 24533 335 06408 854 01636 153 00408 451 00099 762 00023 852 00005 585 00001 281 288 064 014 003 OOI No. of root (n) I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Value of root (x,) ) Min. JO (x) =- Max. No. of root (n) I Value of root (x,) Jo () Max. o (Jn) -Min. 3'8317 0597 0207 5123 7-0155 8666 9815 6I88 10'1734 6813 5062 7221 13-3236 9193 6314 2231 I6'4706 3005 0877 6328 19'6158 5851 0468 2420 22'7600 8438 0592 7719 25-9036 7208 7618 3826 29-o468 2853 4016 8551 32'1896 7991 0974 4036 35'3323 0755 oo0083 8651 38'4747 6623 4771 6151 4I61I70 9421 2814 4509 44'7593 1899 7652 8217 47'9014 6088 7185 4471 51'0435 3518 357I 5095 54'I855 5364 Io6i 3205 57'3275 2543 790I OI07 60o4694 5784 5347 4916 63-6113 5669 8481 2326 66-7532 2673 4098 4934 69-8950 7183 7495 7740 73'0368 9522 5573 8348 76'1786 9958 4641 4576 79'3204 8717 5476 2994 - 0'4027 5939 5702 5547 +03001 1575 2526 1326 -0'2497 0487 7057 8259 + 02183 5940 7247 8730 - o1964 6537 I468 6572 +o800oo 6337 5344 3156 - -I671 8460 0473 8I8o +0'1567 2498 6252 8622 - -1480 II Io 9972 7775 +0-1406 0579 8193 1148 - 0'I342 1124 03IO 0007 +0-1286 1662 2072 0700 - oI236 6796 0769 837I +0oI192 498I 2010 6895 - -I152 7369 4I20 i68o +o'III6 7049 6859 2113 - oIo83 8534 8943 6825 +0'1053 7405 5395 2352 - I1026 0056 7103 3972 +0oI000 3514 6811 5233 -0oo976 530I 5783 1733 + o0954 3333 9020 5353 - o0933 5845 3290 4550 +0-0914 1327 2155 9213 - oo895 8482 1964 8557 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 82'4622 599I 4373 5565 85-6040 1943 6350 2310 88-7457 6714 4926 3069 91'8875 0425 1694 9853 95'0292 3180 8044 6953 98-1709 5073 0790 7820 IOI03126 6182 3038 7301 104'4543 6579 1282 7601 107'5960 6325 9509 1722 110'7377 5478 0899 2151 113 8794 4084 7594 9981 i 7'02II 2189 8892 4250 120-1627 9832 8149 0038 I23-3044 7048 8635 7180 I26-4461 3869 85i6 5957 I29'5878 0324 5I03 9968 132'7294 6438 8509 6159 135-8711 2236 4789 ooo6 I39-0127 7738 8659 7042 142'I544 2965 5859 0290 145'2960 7934 5195 9072 148'4377 2662 0342 2304 151'5793 7163 1401 4280 I54'72IO 1451 6285 9535 157'8626 5540 1930 2978 +0-0878 6187 6039 4105 - o-0862 3466 3413 2884 +0-0846 9463 4803 7192 -0-0832 3427 2981 9746 +o'o8i8 4693 7926 4857 - 0-0805 2673 9448 4029 +0o0792 6843 1724 5187 - 0-0780 6732 5407 9485 +o-0769 I921 3961 3909 - 00758 2031 I569 1671 + -0747 6720 0537 0746 - 00737 5678 6512 8573 +0-0727 8626 0189 2388 - 0-0718 5306 4408 8473 +00o709 5486 5793 0974 - 0-0700 8953 0177 2614 +o-o692 5510 I263 7661 - o-684 4978 2005 1879 +o-o676 719I 8315 5457 - oo669 I998 4772 3973 +o'o66I 9257 2028 7533 - oo654 8837 5698 2572 +o-o648 o6I8 6514 098I - oo64I 4488 1592 6670 +0o-635 0341 6658 3216 Cl02 0= " 02l 0,cN 02 0 02 TABLE IV. 281 Jo (x i) = X- Yi 0'0 0'2 0-4 o-6 o-8 I'O 1'2 I'4 r6 I'8 2'0 2'2 2'4 2'6 2-8 3-0 3'2 3'4 3-6 3'8 4'0 4-2 4'4 4-6 4'8 5'o 5'2 5'4 5'6 5-8 6o x + IO000000000 + 0999975000 + o-999600004 + 0997975 I 14 +0-993601138 +0'984381781 + 0967629156 + '940075057 + 0897891139 + 0836721794 +0'751734183 +0'637690457 + o489047772 + 0300092090 +o-o65112108 - 0-221380250 - 0'564376430 - 0-968038995 - '435305322 -I'967423273 - 2563416557 - 3219479832 - 3928306622 - 4678356937 -5'453076175 - 6'230082479 - 6'980346403 - 7667394351 - 8246575962 - 8-664445263 - 8'858315966 y Nil + oo009999972 + 0039998222 + oo89979750 +0-159886230 + 0249566040 + 0358704420 + 0486733934 +o 632725677 +0'795261955 +0'972291627 + II60969944 + 1357485476 +I'556877774 +1-752850564 +1'937586785 + 2'101573388 + 2233445750 +2'319863655 2'34543306I - 2-292690323 + 2I42167987 + 1872563796 + 1461036836 + 0883656854 + 0 16034382 - 0o865839727 - 2084516693 - 3'559746593 - 5'306844640 - 7334746541 __ __ 282 TABLE V. x 11 (x) x 1(a) $' I1 () x IW(x) oo Nil '45 0'230743570 '90 0497126448 1-35 0'840904230 01 0-005000063 '46 o0236137373 '9I o'503751599 I '36 0849809949 '02 0-010000500 47 o024548938 *92 0'510414946 I37 0858780872 '03 0015001687 '48 0246978674 '93 0-517117001I 138 0867817710 '04 0020004000 '49 0-252426993 94 0-523858282 1-39 0-87692II72 '05 0025007814 '50 0-257894304 95 o0530639310 I140 o 88609198 *o6 0-030013502 '5I 0-263381026 *96 0-537460608 I'4I 0'895330860 07 o0035021441 '52 o0268887571 '97 0o544322705 I'42 0o904638540 *o8 0-040032009 53 0-274414358 '98 o0551226129 I143 o'9I40o5758 09 0'045045577 54 0'279961803 99 0'558171417 I'44 0923463255 IO 0'050062526 55 0285530329 I'0oo0 0565159104 45 0932981780 '*I 0-055083230 '56 0291120360 I'OI 0-572189733 I 46 0942572087 '*2 0-060108065 57 029673238 I 02 0-579263847 1-47 0'952234935 *13 0o065137410 '58 0-302366629I 103 0586381997 1-48 0o96i97I092 '14 0070171639 '59 o0308023722 104 593544734 1-49 0'971781330 '15 0-075211135 '6o 0'3I3704026 105 0o 600752614 1-50 0o981666428 * I6 0080256272 *6i 0'319407973 I -o6 0o608006196 I'5x 0'991627170 '17 0o085307432 -62 0325135997 I '07 0'615306043 152 Ioo00166435I 'i8 0o090364993 63 0-330888532 I 'o8 o0622652724 1-53 I'011778765 'I9 0'095429332 '64 0'336666018 I'09 0630046810 1I54 1-021971216 *20 0-100500834 -65 0o342468895 I'O o637488876 1-55 1-032242518 *21 0-I05579878 *66 0348297605 ' ii o'644979503 I156 1-042593488 '22 0o110666843 *67 0354152590 I'2 0-652519270 I 57 I'053024951 *23 0'115762116 '68 0o360034297 I'I3 0'660108769 '158 i'o63537735 '24 0-120866075 '69 0365943176 I '14 667748588 1-59 I-074132681 '25 0-125979109 '70 0-371879677 I'I5 o0675439326 I 6o0 o84810635 ~26 01311I01599 '71 0377844255 Ii6 0683181582 I '6 I1095572447 '27 0'I36233930 '72 0-383837364 I'I7 0o690975960 I 62 1106418977 *28 0141376489 73 0-38985946I i I8 0o698823068 I 63 1-117351091 *29 0-146529663 74 0o395911007 I'9 0706723524 1'64 I'128369664 '30 0-51693840 '75 0401992463 1-20 0-714677942 I'65 I'I39475574 *31 0-156869409 '76 0'408104296 I'2I 0-722686944 I 66 1*150669712 '32 0o'62056756 77 0-44246975 122 073075II60 I 67 1161952973 '33 o'167256278 '78 42042097I I23 0-738871219 I'68 I'17332626I '34 0-172468361 '79 0426626755 124 0'747047758 1'69 I1-84790486 '35 0'177693400 80 0'-432864802 1-25 0-755281420 1'70 I-i96346565 *36 0-182931789 *8i 0'439135593 I126 0-763572846 I'7I 1-207995429 '37 O'I88I83922 '82 0'445439607 I '27 0-771922691 I 72 I 219738009 '38 0-I93450196 '83 0'45777329 I-28 078033i610 I'73 I'231575249 '39 0'198731008 84 0458149245 I-29 0'788800263 I'74 I'243508096 '40 0-204026756 '85 0464555845 I'30 0797329314 I'75 I'2555375I3 '4I 0-209337840 '86 0-470997619 I'3 o8059I9438 I'76 I'267664463 '42 0-214664660 -87 0-477475069 I32 0-81457I307 I77 I1279889923 '43 0-220007618 -88 0-483988688 1'33 0-823285603 1-78 1-292214874 '44 o0225367121 '89 0-490538979 1 '34 o0832063015 1'79 I1304640310 TABLE V. (continued). 283 7r - x i 80 I'8I 1-82 I 83 i-84 I 85 i 86 1'87 I-88 -89 I -90 I 91 1'92 1'93 1'94 1'95 I'96 1'97 I-98 1'99 2'00 2'01 2'02 2-04 2'05 2'06 2'07 2'08 2'09 2'10 2'11 2'II 2'12 2'13 2-14 2'15 2'I6 2'17 2-I8 2-19 2'20 2'21 2'22 2'23 2-24 11 I (x) 1'317167230 1'329796644 1'342529568 1I355367027 1'3683I0061 1'381359709 1'394517026 1-407783076 1-421158927 1'434645663 1'448244373 1'461956157 1'475782125 I'489723395 1'503781096 I'517956370 1'532250362 I'546664233 1-561199148 I'575856293 1-590636855 1'605542033 I-620573039 I-63573I095 1'651017434 1'666433299 i-68I979944 1-697658635 I'713470648 1'729417273 I'7454998IO I-761719567 1-778077871 1'794576055 1'811215465 182799746I 1'-844923415 I-86I994709 I'879212738 I-896578912 1'914094651 1-931761388 1'949580572 I 967553660 1 985682127 I 1 2-25 2-26 2'27 2'28 2'29 2'30 2'31 2'32 2'33 2'34 2'35 2'36 2'37 2'38 2'39 2-40 2'41 2'42 IW (x 2-003967457 2-022411151 2'041014722 2'059779695 2'078707611 2-097800028 2-II7058510 2*136484642 2'156080021 2'175846257 2'195784977 2'215897825 2-236186453 2-256652534 2'277297753 2-298123813 2'319132429 2'340325336 I I r I iI I 2-43 2'44 t 2'361704281 2-38327I029 2'45 2-46 2'47 2'48 2'49 2'50 2'51 2-52 2'53 2'54 2'55 2'56 2'57 2-58 2'59 2-60 2'61 2-62 2-63 2-64 2-405027363 2-426975075 2-449115981 2'471451912 2'493984712 2'516716246 2-539648394 2-562783055 2-586122143 2-609667592 2-633421351 2'657385389 2-681561694 2'705952269 2'730559137 2'755384341 2-780429941 2-805698017 2-831190666 2'856910009 x 2'70 2'71 2'72 2-73 2'74 2'75 2-76 2'77 2'78 2-79 2-80 2-81 2-82 2-83 2-84 2-85 2-86 2-87 2-88 2-89 2-90 2-91 2-92 2'93 2'94 2'95 2-96 2'97 2-98 2'99 3-00 3'01 3'02 3*03 3'04 3'05 3-o6 3-07 3'o8 3'09 3'10 3'12 31I3 314 3'016107694 3'043474850 3'071086362 3-098944528 3'127051673 3'155410139 3-184022290 3-212890513 3'242017219 3'271404837 3'301055823 3'330972651 3'361157821 3'391613857 3'422343306 3 453348735 3'484632737 3'516197933 3'548046962 3-580182492 3-612607212 3-645323840 3'678335120 3'711643814 3'745252718 3-779164648 3'813382452 3-847908999 3-882747188 3'917899943 3'953370217 3'989160991 4'025275271 4-061716094 4-098486520 4'135589648 4'173028594 4-210806510 4'248926577 4-287392003 4'326206027 4'365371921 4'404892984 4'444772545 4'485013970 - x 3'15 3'-i6 3'17 3'I8 3'I9 3'20 3'21 3'22 3'23 3'24 3'25 3'26 3'27 3'28 3'29 I I 3'30 3.3' 3'32 3'33 3'34 3'35 3'36 3'37 3'38 3'39 3-40 3'41 3-42 3-43 3'44 3'45 3'46 3-47 3'48 3'49 3'50 3'5I 3'52 3'53 3'54 3'55 3'56 3'57 3'58 3'59 I,(x) 4-525620649 4-566596009 4'607943508 4'649666635 4-691768912 4'734253895 4'777125171 4-820386363 4'864041126 4'908093153 4'952546I65 4'997403925 5 '042670227 5'o88348897 5'134443807 5-I80958856 5'227897983 5'275265168 5'323064420 5 '371299790 5'419975369 5'469095281 5'518663697 5'568684817 5 619162888 5 670102192 5'721507056 5'77338I845 5-825730963 5 878558859 5 931870019 5'985668980 6'o39960312 6'094748632 6'150038601 6-205834922 6'262142346 6'318965664 6'376309712 6'434179377 6'492579585 6'551515315 6-610991589 6'671013473 6-731586089 I 2'65 2-882858180 2'66 2'909037340 2-67 2'935449665 2-68 2'962097349 2-69 2'988982613 I I 284 TABLE V. (continued). x 11 (X) x 11(x) x 11( ) x I(x) I 3'60 3'61 3'62 3'63 3'64 3-65 3'66 3'67 3'68 3'69 3'70 3'71 3'72 3'73 3'74 3-75 3-76 3'77 3'78 3'79 3-8o 3'81 3-82 3'83 3'84 3'85 3'86 3'87 3'88 3'89 3'90 3'91 3'92 3'93 3'94 3'95 3'96 3'97 3'98 3'99 6'792714601 6'854404223 6'916660219 6'979487901 7-042892632 7'106879825 7'I71454946 7'2366235Io 7'302391084 7-368763288 7'435745797 7'503344337 7'571564687 7-640412684 7'709894216 7'780015230 7'850781728 7'922199767 7'994275465 8-067014991 8'140424579 8-214510518 8-289279159 8-364736907 8-440890236 8'517745677 8'595309818 8-673589318 8'752590893 8-832321322 8-9I278745I 8'993996193 9-075954517 9-158669467 9-242148147 9'326397737 9'411425473 9-497238668 9'583844704 9-671251025 I I I 400oo 4'oi 4'02 4'03 4'04 4'05 4-o6 4'o7 4.08 4'o9 4'11 4.12 4'13 4'14 4'15 4'1i6 4'17 4'I9 4-20 4-21 4-22 4'23 4'24 4'25 4.26 4'27 4-28 4'29 4'30 4'31 4'32 4'33 4'34 4'35 4'36 4'37 4'38 9'759465154 9-848494681 9'938347267 10-029030650 10-120552634 I0-212921103 10-306144016 I0400229397 10'495185359 I0-591020085 10'687741837 10'785358956 o10883879856 10-983313038 II-08366708I 11'184950646 11-287172471 11'390341384 II-494466292 II1599556I84 II-705620143 11-812667328 11920706992 12-029748470 12-13980II91 I2-250874666 12-362978507 12'476122406 12-590316150 12-705569622 12-821892796 12'939295743 13-057788626 I3'177381705 I3'298085340 13-419909985 13-542866196 13-666964630 13-792216043 4'40 4'41 4'42 4'43 4'44 4'45 4-46 4'47 4'48 4'49 4'50 4'51 4'52 4'53 4'54 4'55 4'56 4'57 4-58 4-59 4-60 4'61 4.62 4'63 4'64 4'65 4'66 4'67 4-68 4'69 4'70 4'71 4'72 4'73 4'74 4-75 4'76 4'77 4-78 4-79 I 14-046221338 I4'I74997247 14'304970189 I4'436151440 14'568552384 14'702184510 14-837059420 14-973188822 15110584538 15'249258499 15'389222754 15'530489464 15-673070904 15-816979464 15-962227657 16-108828111 16'256793575 I6-406136918 16'556871133 16'709009334 16-862564762 17'017550780 17'173980885 17-331868690 17'491227953 17-652072549 17-814416491 17'978273926 I8-143659128 I8'3I0586520 18'479070647 I8-649126207 18-820768025 18'994011070 19'168870460 I9'345361448 19'523499439 19'703299977 19-884778763 20-067951638 I I i ii I 4'80 4-8I 4'82 4'83 4-84 4'85 4-86 4'87 4-88 4'89 4'90 4'91 4'92 4'93 4'94 4'95 4'96 4'97 4'98 4'99 5-00 5'01 5-02 5'03 5'04 5'o5 5-o6 5-07 5-o8 5'09 5-10 20-252834600 20-439443796 20-627795525 20-817906249 21'009792573 21'203471276 21 398859282 21596273684 2I'79543I735 21-996450853 22-199348620 22-404I42793 22-610851286 22-8I9492I89 23-030083764 23'242644448 23-457192854 23-673747769 23-892328160 24'112953174 24'335642142 24'560414578 24-787290180 25-oi6288837 25'247430624 25 480735808 25-716224854 25 9539I8413 26'193837336 26-436002675 26-680435680 439 13-918631291 l TABLE VI. 285 x I, X I (x) I2(x) 0'0 o o 0'2 0'4 o-6 o-8 I'O 1'2 I'4 i 6 I 8 2'0 2'2 2'4 2-6 2-8 3'0 3'2 3'4 3'6 3-8 4'0 4'2 4'4 4'6 4-8 5'0 5'2 5'4 5-6 5'8 6-o I 00000000000 I101002502780 1'04040178223 1I09204536432 1'16651492287 1I26606587775 I'393725584I3 1'55339509973 1'74998063974 '98955935662 2-27958530233 2-62914286357 3-04925665799 3-55326890424 4'15729770350 4-88079258586 5'74720718718 6-78481316043 8-02768454705 9'51688802610 II-301921952I 13'4424561633 16-0104355250 19'0926234795 22'7936779931 27-2398718236 32'5835927106 39'0087877856 46'7375512926 56-0380968926 67'2344069764 Nil '100500834028 '204026755734 '313704025606 '432864802620 '565159103990 '714677941552 -8860919814I5 1 o848io63513 I'31716723040 1'59063685463 1'91409465059 2-29812381254 2'75538434051 3'30105582264 3'95337021738 4'73425389471 5-67010219264 6-79271460136 8-14042457894 9'75946515371 11-7056201430 14'0462213375 16-8625647618 20-2528346003 24'3356421424 29-2543098818 35'1820585061 42-3282880326 50'9461849787 61'3419367775 Nil -0250I668751391 '0202680035615 '0463652789678 '0843529163180 'I35747669767 '202595681546 '287549411997 '393967345826 '526040211741 '688948447698 '889056817580 1'13415348087 1'43374248847 1'79940068733 2'24521244092 2-78829850299 3'44945892947 4'25395421296 5'23245403722 6-42218937528 7'86835133327 9-62578946244 11I7610735829 14'3549969097 17'5056149666 21I3319350638 25'9783957463 31'6203055668 38-4704468999 46'7870947172 286 TABLE VI. (continued). 3 () I4 () I5 (x) 0'0 0'2 o'4 o-6 0'8 I'O I'2 I'4 '6 I 8 2'0 2'2 2'4 2'6 2-8 3'0 3'2 3'4 3'6 3'8 4'0 4'2 4'4 4'6 4'8 5'0 5'2 5'4 5-6 5-8 6'0 Nil '03167083750232 '02134672011869 '024602I6582095 '01110022I0296 '0221684249243 '0393590030648 'o645222328531 'o998922705633 'I48188982086 '212739959240 '297627709533 *407868011092 '549626665935 '730483412160 '959753629490 1-24888076598 I'61191521679 2-06609880918 2'63257822397 3'33727577842 4'21195220660 5'29550364442 6'63554425495 8'29033717554 Io03311501691 12'8451290635 15-9388023977 19'7423554848 24'4148422891 30'I505402994 Nil '05417500694777 '04672017811684 03343620758320 '02I10125859602 o02273712022I04 'o2580066622I87 '0110255569122 'OI93713312I35 '032076938122I '0507285699791 '0773448824914 '114483453137 '165373259392 '234079089848 '325705181936 '446647066782 '604902664549 -810456197666 '07575157832 1'41627570765 1'85127675241 2'40464812914 3'10601585905 3'99207544030 5'10823476364 6'51063229818 8*26861530445 10'4677818331 I3'2I37134973 16'6365544178 Nil '07834723214702 -05268449532285 '04205557100196 '04876350693866 '03271463155956 -03687894919051.02151905049781 'o2303561449592 -02562481265409 -o29825679323I2 o0163735913822 '0262565006355 '0407858678054 '0616860125932 '0912064776610 '132263099020 '188614829615 '265085036586 '367838059088 '504724363113 '685710773430 '923416136884 1'23377754356 I163687810838 2'I5797454732 2-82877168171 3-68900194663 4'78838143757 6 18903056865 7'96846774238 TABLE VI. (continued). 287 x 16x(x) I7(x) I ) _.x. 0'0 0'2 0'4 o-6 o-8 I'0 1'2 1'4 I 6 I 8 2'0 2'2 2'4 2-6 2-8 3'0 3'2 3'4 3'6 3'8 4'0 4'2 4'4 4'6 4'8 5'0 5'2 5'4 5'6 5'8 Nil '08139087425642 '07893980971214 '05I02559I32723 ~05582022868887 *04224886614771 *04682085631142 *03175196213558 *03398740613950 '03827978932673 '02I60017336352 '022919467II786 0o2508I36715570 0o2850453706344 *0137719020155 '0216835897328 '0333248823452 '0501531656813 *0741088738166 *10775668598I 1 54464799871 *218632053769 '305975090770 '423890764347 '581912714514 *792285668997 I'07068675643 I'43713021810 I'917o0069457 2'54297113760 3'35577484714 Nil *1OI198660852I19 '08255240920874 *07438834749717 '06331639053615 '05159921823120 o05580928790861 *04173686673046 '04450598913012 '03104953102941 '03224639142001 '03449225284743 '03849664857007 '02I534I5828I86 '02266357538382 '024472 I872992 '02729479022559 '0116036566222 '0180554571973 '0275537875687 '0413299635012 '0610477626605 '0889386166028 'I279755496I4 *182096322090 '256488941728 '357956089960 '495379239735 '680308520630 '927710973612 Nil.0I224829I584037 o'0~637748154995 o08164357788982 '07165452506106 'o7996062403333 '06433537513798 '05150954051219 '05446656506452 '04116770209099 *04276993695123 '04607607604085 '03124988823159 '03243684776533 '03454025096400 '03813702326455 '02141017510822 '022373403II95I 0o2389320693838 '02624273178058 '0298099276I666 '0151395I15677 '0229885833970 '0343999611745 '0507984417519 '0741166321596 'o06958821921 '152813670643 '216329392995 '303668787505 '422966068203 6-o I'256918o48n 288 TABLE VI. (continued). x Ig(z) I0 (X) III(x) 0'0 0'2 0'4 o'6 o-8 I'O0 I'2 I'4 '6 I ' 2'0 2'2 2-4 2'6 2'8 3'0 3'2 3'4 3-6 3'8 4'0 4'2 4'4 4'6 4'8 5'0 5'2 5'4 5'6 5'8 6'0 Nil 'oI4275848890728 '0oI416588756oo '0'o5473I2431307 '09734041402172 o08551838586274 '07287877246335 '06116775736690 0o6394240656000 '05115736151949 '05304418590271 '05732884540826 *04164060359505 04345596570386 '04691462615510 '03132372988831 '03243914684482 *03434700765661 '03752315248879 '02126860I12417 '02209025303452 '02337343287863 'o2534376788633 'o2832351074598 I0127681829170 0193157188168 '0288520225117 '0425979933861 '0622245406441 0900ooo39735967 '129008532906 Nil 'oi6275823817735 '0'32832I4795I93 'oI"6405959o224 oI00293I906I7555 '09275294803983 'o8I72I64429560.088138I8331745 '07313576845I53 '06103405714922.0630I696387935 '06797479795484 '05I94355352977 '0o442561241924 '0595I34I501153 '04194643934705 '04381550080109 '04720461248306 '03131630693989 '03233568560836 'o3403788961327 '03681942087915 '02112771477116 'o2182970173372 '02291775581322 '02458004441917 'o2708643630312 '0108203593556 '0163219409248 '0243461108260 '0359404694846 Nil 'oI825072993174 '*o5514780037287 '013447130560011 o'0"06485828421 '0o1~24897830849 '0'o9365304020 '0951597501248 '0822695995590 'o884091314720 '07272220233597 '0779029085679 '0620975653574.065I648458289 '05II932971830 '0526103656940 '05544588441373 '04109000313633 '04210336156069 '0439292909243 '04713082278832 '03126089602839 '03217791653765 '0336828581676 '0361086702855 '03995541140110 0o2159649826893 '02252258836536 0*2393189448412 '02605186730033 '02920696795753 BIBLIOGRAPHY. (Other references will be found in the text.) I. TREATISES. RIEMANN. Partielle Differentialgleichungen. NEUMANN (C.). Theorie der Bessel'schen Functionen.,, Ueber die nach Kreis-, Kugel-, und Cylinderfunctionen fortschreitenden Entwickelungen. LOMMEL (E.). Studien fiber die Bessel'schen Functionen. HEINE (E.). Handbuch der Kugelfunctionen. TODHUNTUER (I.). Treatise on Laplace's, Lame's, and Bessel's Functions. BYERLEY. Treatise on Fourier Series and Spherical, Cylindrical, and Ellipsoidal Harmonics. FOURIER. Theorie Analytique de la Chaleur. RAYLEIGH. Theory of Sound. BASSET. Treatise on Hydrodynamics, Vol. II. II. MEMOIRS. BERNOULLI (Dan.). Theoremata de oscillationibus corporum filo flexili connexorum, etc. (Comm. Ac. Petrop. vi. 108.),,,, Demonstratio theorematum, etc. (ibid. vI. 162). EULER. (Ibid. vII. 99 and Acta Acad. Petrop. v. pt. I. pp. 157, 178.) PoIssoN. Sur la distribution de la Chaleur dans les corps solides (Journ. de l'fEcole Polyt. cap. 19). BESSEL. Untersuchung des Theils der planetarischen St6rungen, etc. (Berlin Abh. 1824). JACOBI. Formula transformationis integralium definitorum, etc. (Crelle xv. 13). HANSEN. Ermittelung der absoluten St6rungen in Ellipsen, etc. pt. i. (Schriften der Sternwarte Seeburg (Gotha, 1843)). HAMILTON (W. R.). On Fluctuating Functions (Irish Acad. Trans. xIx. (1843) p. 264). HARGREAVES. On a general method of Integration, etc. (Phil. Trans. 1848). G. M. 19 290 BIBLIOGRAPHY. ANGER. Untersuchungen fiber die Function 71, etc. (Danzig, 1855). SCHLOMILCH. Ueber die Bessel'sche Function (Zeitsch. d. Math. u. Phys. ii. 137). RIEMANN. Ueber die Nobili'schen Farbenringe (Pogg. Ann. xcv.). LIPSCHITZ. Ueber die Bessel'sche Transcendente I. (Crelle LVI.). WEBER (H.). Ueber einige bestimmte Integrale. (Crelle LXIX.)., Ueber die Bessel'schen Functionen u. ihre Anwendung, etc. (ibid. Lxxv.).,, Ueber eine Darstellung willkiirlicher Functionen durch Bessel'sche Functionen (Math. Ann. vi.). Ueber die stationaren Stromungen der Elektricitat in Cylindern (ibid. LXXVI.). HEINE (E.). Die Fourier-Bessel'sche Function (Crelle LXIX.). HANKEL (H.). Die Cylinderfunctionen erster u. zweiter Art (Math. Ann. I.).,, Bestimmte Integrale mit Cylinderfunctionen (ibid. vIII.).,, Die Fourier'schen Reihen u. Integrale fur Cylinderfunctionen (ibid. vIII.). NEUMANN (C.). Entwickelung einer Function nach Quadraten u. Produkten der Fourier-Bessel'schen Functionen (Leipzig Ber. 1869). LOMMEL. Integration der Gleichung Xm+ d2m+1 y x 2 dx2M+l + y = 0 durch Bessel'sche Functionen (Math. Ann. II.).,, Zur Theorie der Bessel'schen Functionen (ibid. III., IV.).,, Ueber eine mit den Bessel'schen Functionen verwandte Function (ibid. IX.).,, Zur Theorie der Bessel'schen Functionen (ibid. xiv., xvI.).,, Ueber die Anwendung der Bessel'schen Functionen in der Theorie der Beugung (Zeitsch. d. Math. u. Phys. xv.). SCHLAFLI. Einige Bemerkungen zu Herrn Neumann's Untersuchungen fiber die Bessel'schen Functionen (Math. Ann. III.).,, Ueber die Convergenz der Entwickelung einer arbitraren Function zweier Variabeln nach den Bessel'schen Functionen, etc. (Math. Ann. x.).,, Sopra un Teorema di Jacobi, etc. (Brioschi Ann. (2) v.).,, Sull' uso delle linee lungo le quali il valore assoluto di una funzione e costante (Ann. di Mat. (2) vi.). DU BOIS-REYMOND (P.). Die Fourier'schen Integrale u. Formeln (Math. Ann. iv.). BIBLIOGRAPHY. 291 MEHLER. Ueber die Darstellung einer willkiirlichen Function zweier Variabeln durch Cylinderfunctionen (Math. Ann. v.). Notiz fiber die Functionen Pn (cos 0) und J (x) (ibid.). SONINE (N.). Recherches sur les fonctions cylindriques, etc. (Math. Ann. xvI.). GEGENBAUER (L.). Various papers in the Wiener Berichte, Vols. LXV., LXVI., LXIX., LXXII., LXXIV., XCV. STRUTT [Lord Rayleigh]. Notes on Bessel Functions (Phil. Mag. 1872). RAYLEIGH. On the relation between the functions of Laplace and Bessel (Proc. L. M. S. ix.). POCHHAMMER. Ueber die Fortpflanzungsgeschwindigkeit kleiner Schwingungen, etc. (Crelle LXXXI.). GLAISHER (J. W. L.). On Riccati's Equation (Phil. Trans. 1881). GREENHILL. On Riccati's Equation and Bessel's Equation (Quart. Journ. of Math. xvI.).,, On Height consistent with Stability (Camb. Phil. Soc. Proc. iv. 1881). BRYAN. Stability of a Plane Plate, etc. (Proc. L. M. S. xxII.). CHREE. The equations of an isotropic elastic solid in polar and cylindrical coordinates (Proc. Camb. Phil. Soc. xiv. p. 250). HOBSON. Systems of Spherical Harmonics (Proc. L. M. S. xxII. p. 431).,, On Bessel's Functions, and relations connecting them with Hyperspherical and Spherical Harmonics (ibid. xxv. p. 49). THOMSON (J. J.). Electrical Oscillations in Cylindrical Conductors (Proc. L. M. S. xvII. p. 310). McMAHON (J.). On the roots of the Bessel and certain related functions (Annals of Math., Jan. 1895). 0 8 6 -a \ I. 1.2 I 6 18 2 0 2 2 2 2 6 2 8 2 8 2234363 04 2446464850502 23 56 56 8606 2646666 68 0 72 7 76 788082848 6?0 9 2 94 86 88 - 18818 0418.~6 1088118011 2 11.411 18128018821.612.61.81.0 14 611811 21,- E- 5 0 E2 - 1-214 tj U2 I ~ ~ ~ ~ ~ ~ 21. 1018 4 1 GRAPH OF J,, (X) AND) J1 (X). The abscissae represent arguments and the ordinates values of the functions on the scales indicated by the numbers on the graph. Iw ~3 ---- ^ --- —-~ 1 ^i C I-IiI IIIIIU-f — IGLI ^ 0* 0^ 061 101 141 f 201 241. 281 OK^ 341. 381. 421. 46V 81 &~~. ~* 661- 601- 6*' 6kj 70 1 21. 76li T- E. ~. ~. 881- S*^ ' 961 82 | ~!1^ --- —"^ ^ ---- - -- -- j tr |d GRP F o^NDJ() Th bcsa ersn ruet n h riae ale ftefntoso h clsidctdb h 4 ~ ~ ~ ~ ~ ~~~~~~ubr ntegah