1915-16.] On Systems of Partial Differential Equations, Etc. 303 
obtained by A. H. Leahy* and Ad. Schmidt,]* but the fact that the 
coefficients occurring in the formulae can be regarded as wave-functions 
does not appear to have been noticed previously. 
We easily find that 
X cos o> + Y sin co + PL = a[(£ 2 - rf) cos co + 2^ sin w] + /3[(£ 2 — rjfj) sin co - 2 £rj cos co] 
iiyiP + r, 2 ), 
where 
£ = x cos io + y sin co + iz, rj — x sin co — y cos co + is. 
Let us now write 
a = r sin 6 cos (3 = r sin 6 sin cf>, y — r cos 0 , 
and make use of Jacobi’s expansion j 
(y + ia cos i/r + i/3 sin i f/) n = 
n\ 
(cos 0)e 
~ n ( w + m ). ! 
Then if we write £ = or cos v, y = cr sin v, we find that 
m=+n 
(Z — 2 -X cos co — iY sin co) w = 2, H » (*, y, «> s)r"P n (cos 6)e 
(7> 
where 
HT-(-l)V 
= ( 1 
ft! 
m; 
_2n e ^( w _2,) 
(w + m) ! 
e mco[( x _ ^) e ™> + ^ _j_ / s )] w ™[(# + iy)e- i0i z(2 — is)] n+rfl . 
Introducing polar co-ordinates X = R sin = O' cos (p Y == R sin 0' sin <f> 
Z = Rcos O', and expanding the left-hand side of (7) in powers of we get 
m= +n 
R W P W (cos 6')e~ ik V = ^ L n ’ m (x, y , 2 , s)r n P™(cos 0)e~ im( t> 
where L n is the coefficient of in the expansion of the function 
. ( n + k ) ! 
( — 1 ) n i m+k p ~ n + rjl y e im0i [(x — iy)e ioi + i(z + isff 1 m \(x + iy)e~ i(a + i(z - fs)] w+m 
in ascending and descending powers of e iai . This coefficient is evidently a 
polynomial of degree 2 n in x, y, z, s, and satisfies the equation of wave- 
motion ; it may, in fact, be regarded as a standard homogeneous polynomial 
solution of this equation. § 
* Proc. Roy. Soc. London , vol. lvi (1894), p. 46 ; “Papers printed to commemorate the 
Incorporation of the University College of Sheffield” (1897), pp. 60-88. The formula which 
I have used recently (Terrestrial Magnetism , Sept. 1915) for the mean value of a function 
round a circle can be deduced immediately from a result given in the first of these papers. 
t Zeitschr. fur Math. u. Phys., Bd. xliv (1899), p. 327. 
I Of. E. W. Hobson, Encyclopaedia Britannica , vol. xxv, p. 651. 
§ See the author’s Electrical and Optical Wave Motion , p. 112. 
