Conway—The Partial Differential Equations of Mathematical Physics. 198 
Let us assume for # the form 
w+ Ay, + Any +... 
when , %.. . are functions of 7 — ¢, such that 
Wi=% Woah, Pr=h..., 
the accents denoting differentiations, and A,, A, ... are functions of 7. The 
result will be, on writing 
K = 7(n—1)(n—3), 
, K / K \ / 1 K 
(244-5) 4 hi(244 +4 1 en) a fa (244 + As % As) + eee 
Kquating to zero the coefficients of the functions , we get 
ig 1 
Aber oe Ks ee oan rye 1] 
A. % 3, [(2 — 2) i 
Ka Ke 1 i : 
ee = [@=2P —1[e=o9P=81, &e. 
Als 32 198 [ (a P=1][(m ) I, &se@ 
Hence, we get a series for ¢ in descending powers of 7. It terminates if n be 
odd, but if 2 be even, the series does not terminate; but, by taking any member 
to terins, we get an ‘‘asymptotic” solution. When 2 is even, as before, a 
solution is 
i (w) du 
The path of integration which we take now will be a line coming from — « 
passing around the point «w=¢—r and returning to —o, or if we regard 
the w-plane as being a two-sheeted Riemann surface connected along the 
line joining ¢—,7 to ¢+ 7, a line passing from —® in one sheet to — 0 
in the other. We shall call this integral 
Kins) (4, 7); 
and we shall speak of the function / as its generating functior 
We easily find 
1 / 3(n-2) I ; 
IK yy oy) = —__——— ; Coo 
CO) 1 B28 soo @=B) OP ; 
