From this form it can be shown that

Q.G0 =i P.(m) log KÂ±|- W^Cm),

where WÂ»_i(/x) is a rational integral function of degree n â€” I in m; it can be shown that this form is in agreement with the definition
of Qnto by series, for the case mod /*>i. In case mod /n* x . but differs from it by an imaginary multiple of Pâ€ž(m). It will be observed that Q n (i), Q n ( â€” i) are infinite, and Qn(x) =o. The function Wn-i(ju) has been expressed by Christoffel in the form
*

*2 "-?Pâ€ž_ lW + - 2 "-
*

*â€¢w+|^ p â€” w+-
*

*I . n ^' . ' 3 . n â€” I
*

*and it can also be expressed in the form
*

*~P (p.)P^ 1 (p.)+~P l (p)?^. 2 (p.) + . . . +Pâ€ž-,(m)Po(m).
*

*It can easily be shown that the formula (28) is equivalent to
*

*which is analogous to Rodrigue's expression for PnM-
Another expression of a similar character is
*

*It can be shown that under the condition mod \u â€” V (u- â€” 1)\
>mod \ti â€” V (v?â€” 1)|, the function il(n â€” u) can be expanded in the form 2(2Â»-|-i)PÂ«(tt)Qâ€ž(tt) ; this expansion is connected with the definite integral formula for Q n (v) which was used by F. Neumann as a definition of the function QÂ»(p.), this is
*

*which holds for all values of \i which are not real and between Â± I. From Neumann's integral can be deduced the formula
*

*d4>
*

*â€¢â€¢00
Qn(M) = J ,
*

*iM+vc^-o.cosh^r'
*

*which holds for all values of p. which are not real and between
="= 1 , provided the sign of V (/* 2 â€” 1 ) is properly chosen ; when p is
real and greater than 1, V (p 2 â€” 1) has its positive value.
By means of the substitution.
*

*Jm + V(m 2 â€” i).cosh \fr\\ii â€” V(m 2 â€” l).cosh x! = i,
*

*the above integral becomes
*

*Qâ€ž0) = J *V~V (m 2 - 1) cosh xNx. where xa = ^og c j~. '
*

*This formula gives a simple means of calculating Qâ€ž(p) for small
values of n; thus
*

- w-/>-->*eÂ±j.

If, in Legendre's equation, we differentiate m times, we find

. . n d m **u , . . d M+1 u , , ., , , .d m u

â€¢ c 11 1 d m u , m , , ,v m d m u

it follows that u = ^'hence Â«â€ž = 0* 2 -i)- - a -^-

2m'"

The complete solution of (26) is therefore

when m is real and lies between Â± 1 , the two functions

U-M - ) 4u'" ' U_M - ) C^" 1 are called Legendre's associated functions of degree <n, and ordur m, of the fhrt and second kinds respectively. When /j is not real and between =*=i, the same names are given to the two functions

Qi(p) =/Â»XÂ«-V (m 2 - I) -sinh Xo = M-^oSj^~ I.

Neumann's integral affords a means of establishing a relation between successive Q functions, thus

Â»QÂ»-(2rt-i)/Â«QÂ»-i + (n-l)(?Â»_2

"1 Â«P,(tt) + (n- i)Pâ€ž- 2 (h.) - (2w- p â€” u

pP n -i{u)du

-ITJ

= -|J _,(2Â»- l)PÂ»-l(") =0.

Again, it may similarly be proved that

19. Legendre Associated Functions. â€” Returning to the equation (26) satisfied by Â«â„¢ the factor in the normal forms â€ž_, â„¢ mtfr.uâ„¢,

we shall consider the case in which n, m are positive integers, and

n^m. Let Â« = (m 2 â€” i)* m v, then it will be found that v satisfies the equation

(m 2 -0

dp'" '

(V-i)

U d m Q*W

dp m

in either case the functions may be denoted by P n (m), Qn (m). It can be shown that, when p is real and between Â± 1

p:m =5^11 (S3 im ^io.-i)"^o.+i)-i

2Â»(n-m)l\l+p) dp" 1 ^ V " +l ' <â€¢ In the same case, we find

Pr +2 (cos 8)-2(m + i) cot $ Pâ„¢ +1 (cos 0)

+ (n â€” m)(n+m + l)Pâ„¢(cos 8)=o, (ra-Â«+2)Pr +2 (cos 0) - (2n+3) M Pr + i(cos 0)

+ (Â»+m + l)Pr(cos0)=o.

20. Bessel's Functions.â€” -If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are a, p, $, the cylindrical co-ordinates; in that case Hi = i, Hi = i, H 3 = i/p, and the equation (25) becomes

d"v yy 1 av 1 a-v d~ 2 + <v +p dp v a<^> 2 ~ '

To find the normal functions which satisfy this equation, we put
V = ZR*, when Z is a function of z only, R of p only, and * of <f>, the
equation then becomes
*

*I ^?j_Â± /^P:j_I ^E\ j.1 I 4!?_
Z dz- + K Up 2 + p dp/ V* d^~Â°-
*

*1 <i 2 Z
That this may be satisfied we must have y~r^ constant, say =k',
*

*tt yj conetant, say = â€” m-, and R, for which we write u, must
satisfy the differential equation
d 2 u , I dti
*

*dp'
*

*. 1 du /,â€ž pi-\
*

*it follows that the normal forms arc e ^ z '-m<p.u(kp), where u(p)
satisfies the equation
*

*",7/ T n ii t n'"\
*

*(29)
*

*d 2 u , I du , / n7 ! \
*

*d?+pd7 + { l -7) u=0 -
*

*This is known as Bessel's equation of order m\ the particular case
*

*d"-u , I du
*

*d9 + pT P + u=0 '
*

*(30)
*

*corresponding to m = o. is known as Bessel's equation.
*

*If we solve the equation (29) in series, we find by the usual process
that it is satisfied by the series
*

*+0
*

*2.2OT+2 r 2.4.2/W-(-2.2W-|-4
*

*the expression
*

*P" \, P 2 ,
*

*2"*II(m) ( 2.2m+2 2.4.2m+2.2m+4
*

*...| s
*

*or
*

*(â€” i)V +2n
*

*/ |) 2"' +2 "n(m+Â»)n(w' l
*

*is denoted by J m (p). """i
When m=o, the solution
*

*i _pJ + _A_...
*

*2 2 ' 2 2 . 4 2
*

*of the equation (30) is denoted by Jo(p) or by J(p).
*