1915-16.] On Systems of Partial Differential Equations, Etc. 305 
We thus have <F = 0, ^ = 7r, p — 1, cr = 0. A comparison of the results leads 
to the formula 
d n 
dc r 
n + m + 1, m - n, m- k + 1 
1 + c 
rn>k 
(_1 )«+m(n + m)\ F ( 
2 m (n - m ) ! (m — k ) ! 
I ol/ ^ ( 1 + c) k ~ m F(n + k+l,k-n,k-m+ 1 
2 k (n-k)\(k-m)V ’ \ 2 / ^ 
which can be verified without difficulty by differentiation. 
In the general case it appears from our results that Leahy’s functions * * * § 
L (n, m, r, p) and K (n, m, r, p) can be expressed in terms of Jacobi’s 
polynomial, *(■ which is defined by the equation 
F (~p,a+p, c, x ) 
X 1 C (1 x) c a —{x c +v-\l^) a+p - c } . ( 8 ) 
c(c+ 1) . . . (c + 'P - 1 ) dx p 
where p is a positive integer. 
§ 3. An interesting property of Jacobi’s polynomial may be deduced 
from the fact that if we write 
p 2 = l-£— 7], cos 2 U = 
i+rj—V 
the equation of wave-motion is satisfied by 
-.m+k 
(^“^"[(l - 0(1 - 77)]“ s - e - ik ®- im *F(m - n, m + n + 1, m — k + 1, g) 
F (m — n, m + n+ 1, m — k+ 1, rj)'. 
Expressing the solution ( x,y,z,s ) in terms of the solutions of the above 
type, and vice versa, we find that there are two equations of the following 
kind : — 
(1 - i~ vY F ( -p,a +p, c, = ^A s ¥(-s,a + s,c, £)F( -s,a + s, c, v ), 
F(-s,a + s,cJ)F(- .S-, a + c, rj) = ^ ff,(l - £ ~ vY F ( ~P,a + P,c, 1 )• 
The first expansion is already known under a slightly different form.§ The 
coefficient A s is given by the equation 
A = (a + 2s) Tff + s)V(p + l)V(p + a - c + l)T(a + g) 
5 V ' } Y (a-c + s+ l)T(p - s + l)T(s + l)T(r)r(p + a + s+l)' 
To determine the coefficient B p , we put rj — 0, then 
* To pass from Leahy’s notation to ours we must put 2 u—p. 
t Orelle’s Journal , Bd. lvi (1859), p. 156 ; WerJce , Bd. vi, p. 191. 
I See the author’s Electrical and Optical Wave Motion , p. 109, Ex. 1. 
§ H. Bateman, Proc. London Math. Soc., ser. 2, vol. iii (1905), p. 123. 
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