J1 (P)  = -dJoP(P), Yi(P) dYo(p) 22 . Bessel's Functions as Coefficients in an Expansion.—It is clear that e'P COS 4' =e'x or e'P sir' 0=e," satisfy the differential equation (31), hence if these exponentials be expanded in series of cosines and sines of multiples of ¢, the coefficients must be Bessel's functions, which it is easy to see are of the first kind To expand e'P sin 4', put e'4' =t, we have then to expand e_P('_,-i) in powers of t . Multiplying together the two absolutely convergent series \ e °7-1ml(2)mtm,e LP` - ( m~m 12p1 tm, we obtain for the coefficient of tm in the product l\ J 4 2mm ! 2.201+2+2.4.2171+2.2m+4- ... } or Jm(P), hence the Bessel's functions were defined by Schlomilch as the coefficients of the powers of t in the expansion of elp(7-i '), and many of the properties of the functions can be deduced from this expansion . By differentiating both sides of (32) with respect to t, and equating the coefficients, of t"'-' on both sides, we find the relation J.-1(p) +Jm+i(P)=2p Jm(P), which connects three consecutive functions . Again, by differentiating both sides of (32) with respect to p, and equating the coefficients of corresponding terms, we find 2dr (P)-m 1(P)-Jmi(P)- In (32), let t=e b, and equate the real and imaginary parts, we have then cos (p sin 0)=Jo(P)+2Jz(p) cos 24)+2J3(P) cos 30+... sin (p sin 0) =2Ji(p) sin ¢+2J3(p) sin 30+ .. , wee obtain expansions of cos (p cos 0), sin (p cos 0), by changing 4) into 1-4 . On comparing these expansions with Fourier's series, we find expressions for J,,,(p) as definite integrals, thus Jo(P) = J u cos (psin ¢)d¢, J,,,(p) _; fcos (p sin 0) cosmrbdetz . (m even) Jm(P) =,r' j sin (p sin ¢) sin m¢d¢ (m odd) . It can easily be deduced that when m is any positive integer i,, (P) = jo cos (m0-p sin 0)1¢ . 23 .

Bessel's Functions as Limits of

Legendre's Functions.—The system of orthogonal surfaces whose parameters are cylindrical co-ordinates may be obtained as a limiting case of those whose parameters are polar co-ordinates, when the centre of the spheres moves off to an indefinite distance from the portion of space which is contemplated . It would therefore be expected that the normal forms e ±(Jm(Xp)s nm4) would be derivable as limits of r nd'IPn (cos 8)s mip, and we shall show that this is actually the case . If 0 be the centre of the spheres, take as new origin a point C on the axis of z, such that OC =a; let P be a point whose polar coordinates are r, 0, 4) referred to 0 as origin, and cylindrical co-ordinates p, z, 4) referred to C as origin; we have p = r sin B, z = r cos 0 -a, hence (a) "P"(cose) = see() (1+0 "P"(cos 0) . Now let 0 move off to an infinite distance from C, so that a becomes ( m-1 (2) .L ( ( ./II(n ni)II(n) 2) z" "=0 (31) or thus we have e'P('-d-1) = Jo (P) +tJ1(P) + . - . +tmJ m (p) + .. . —t 1Ji(n)+ . .+(—I)'"t-mJ,n(P) (32) = F'tmJm(P) infinite, and at the same time let n become infinite in such a way that n/a has a finite value X . Then as n L sec" B=L (sec a) = 1, L (t +a!) =eM' and it remains to find the limiting value of P,(cos 0) . From the series (15), it may be at once proved that Ps(cos 0) = - (n+!) n (sin 2) 2+ .. . 1 (—1)m8(n+mlz.2(m2m+I) (sin 2) zm where S is some number numerically less than unity and m is a fixed finite quantity sufficiently large; on proceeding to the limit, we have LP', (COS np) =I ---+224.1-...+(-I)m8122 42 (2m)2 where Si is less than unity . Hence L Pn (cos?P) = Jo(7sP)• n-a0 n d (- h2) m (-2)mPm ( ) d(PL)m hence L n "'Pn (cos P) =J,(P)• n- so n It may be shown that Yo (p) is obtainable as the limit of Qs (cos n) the zonal harmonic of the second kind ; and that Ym(p) =Ln "'Q: (cos ) .

24 . Definite Integral Solutions of Bessel's Equation.--Bessel's equation of

order m, where m is unrestricted, is satisfied by the expression p'° f e"e'(12- I)m-Idt, where the path of integration is either a curve which is closed on the Riemann's surface on which the integrand is represented, or is taken between limits, at each of which e,Pi(t2-1)TM+f is zero . The equation is also satisfied by the expres- sion f ewP(t - I 1)t TM-1dt where the integral is taken along a closed path as before, or between limits at earn of which e12P(t-t 1)t_m_I vanishes . The following definite integral expressions for Bessel's functions are derivable from these fundamental forms . I i P) m f ~e'P cos ci sin 2"' J m(P)=II(-i)II(m-a) 0 where the real part of m+z is positive . Ym(p)+zai.emn'sec ma.J„(p) m—1—n0 n(1 ) (2) '"f o +k r. ecp cosh 4, sinh 2mOdd) where the real parts of m+i, p are positive; if p is purely imaginary and positive the upper limit may be replaced by co . Ym(p)— ar.en— sec m7r.Jm(p) 2mnil"I(--in) (P) rn /' _e II(- ) 2 J o ;pa sinh 2'4;4 Ln mm has been introduced for them . We denote the two solutions of the equation by Io(r), Ko(r) when Io(r) = Jo(1r) =1+5 + 22'-424- + .. . _! f o cosh (r cos 0)4, and Ko(r) =Yo(or)+212rJo(Ir) =f: e`r cos h+dsp = f 1 cos (r sinh 1/')d# . The particular integral Ko(r) is so chosen that it vanishes when r is real and infinite; it is also represented by a0 cos V J o (v2+r2)dv, (U e-"" J.° I (u2-1)du . The solutions of the equation dr2+s dr (I +5) uo are denoted by Im(r),,( Km(r), where ))} Im(r)=2mII(m) I+2.2m-{-2+2.4.2m+2.2m+4+ ... =(2r)" w„Io(r), )) when m is an integer, and Km(r) _ (2Y)md 2),nKo(r) =e-#lm+r .

Ym(ir)+ZwrJm(4r) 26 . The Asymptotic Series for Bessel's Functions.—It may be shown, by means of definite integral expressions for the Bessel's functions, that j( )} Jm(P) = P P cos (211+5—0 + Q sin (2~+I-P) Ym(P) = . / e "' sec ma P sin (2'r+ -P) -Q cos (2F+ -P) where P and Q denote the series pI _(4m2-12)(4mn2-32) I .2 . (8p)2 + (4m2 - 12) (4m2 - 32) (4m2 - 52) (4m2 - 72) I.2.3.4(8p)' 4m5-12_ (4m2-12)(4m2-32) (4m2-52) Q- I.Bp I.2.3.(8p)3 -~ .. These series for P, Q are divergent unless m is

half an odd integer, but it can be shown that they may be used for calculating the values of the functions, as they have the property that if in the calculation we stop at any term, the error in the value of the function is less than the next term; thus in using the series for calculation, we must stop at a term which is small . In such series the remainder after n terms has a minimum for some value of n, and for greater values of n increases beyond all limits; such series are called semi-convergent or asymptotic . We have as particular cases of such series: Jo(P) I a 12.(-8p32)2+ 11.22.332 ..52.72 = \1,~d2p cos (4 - p) ) I.24(8p)4 .. Again, since we have P ':(cos p) =sin"' Bd"P,(cos 0) d(cos 0)"m dTMPn LnmPn (cos') = (cos!) d2u I du dr2+r dr-=o and by We find also Im(r) -" 1 .3.5 .. (2m— 1) ,Jo cosh (r cos 0) sinzm¢d¢ Km(r) ( - I)mrm f° . 3 .5 . . . (2m -I) o e rcosh4 sinh 2.4,4 I =(-1)"'3.5 ..

.2m-Ir"` au cos u o (u2+r2)n+f u . under the same restrictions as in the last case; if p is a negative imaginary number, we may put so for the upper limit .