272 
ME. E. W. BAENES ON THE THEOEY OF THE 
And therefore if we put 
f„{a) — + oj.) — 2S,,(o) — S,,(a | coj) 
u'e shall have 
f,[a + ojj) —/„(«) = 0. 
Now oS„(«) and S„(«/aj|) are algehraical polynomials of a, and therefore is 
also such a function. And, therefore, since it is simply periodic of period it must 
he a constant. Thus we have 
since 
oS;,(« -)- 0 J. 2 ) — 2 S„(a) = S„((:< jwj) + constant.(2.) 
Again, integrating the relation (l) with respect to a between 0 and Wo, we have 
toi + a)2 rwj roj2 
2S„(«)da — ~ oS„(o)(/« = 0, 
u" Jo" Jn' 
Su{(i\(o..)da = 
J f) 
OJ.y 
h n + l(0 I 
?i + 1 
Integrating the relation (2) in the same manner between 0 and wj, we obtain for 
the value of the constant 
1 / I \ 7 h „ t.|(o I COj) 
And thus is the unique algebraic solution of the equation 
/■(« + <»,) - /(a) = S„(« I <„,) + , 
with the condition/'(o) = 0. 
From the symmetrical nature of the e(piations which give oS„((r|c/jj, ( 0 . 2 ), we see that 
this function itself must be symmetrical in Wj and Wo. 
§ 4. If new we assume 
oS;,(a|c(j,, 0 ) 2 } = a«+o + . . . + apq 
the calculation of the liighest coefficients may he readily effected. 
For we have 
oS,,(a + Wj) — I H“ 
S'„+i(c|a)d 
•/?{+! 
('?/ + 1) COn 
a” , 
a" ‘(D., 
1 / *> "" \ *> / ‘? 
, i / \* ) / O 
n 1 
Hence if we substitute the assumed expansion for equate corresponding 
powers of a, we find 
(it fi- 2) O)^ ^h + 2 — 
(w + 2) {n. + 1) 
1 . 2 
O) 
(li -f- 1) Wo ’ 
a «+.3 + (tt "h 1) a,12.1 — — .f. 
