120 Mr. A. Cay ley on an Analytical Theorem relating to 



tliat, 6 being small, the function in question oc sin''^ 0, or what 



is the same thing, a (1 — cos Oy. In the former part of the 

 memoir, Plana shows that this is not the true form of the cleve- 

 lojimcnt; the foregoing development must therefore be illusory; 

 and Plana in fact shows, by a laborious induction carried as far 

 as My, that all the coefficients IM vanish identically. The iden- 

 tical equation Mi=0, where i is any positive integer whatever, 

 constitutes the analytical theorem above referred to. Plana's 

 expression for the function Mj is as follows: — 



Bp B3, B5, &c. denote Bernoulli's numbers as given by the 

 equation 



(B,= g, B3= gQ, B5= ^, B7= g^, &c. I have, in conformity 



with the usual practice, written the equations so as to make 

 these numbers all positive ; with Plana they are alternately po- 

 sitive and negative). And in the equation 1C3, writing for k 



its value ^ — t, we have, \ being any positive integer, 

 ,-. »NAr, 1 l/i 7N X.^-It, (1+^')^ 



(l+j)XGx=_-^(l-f^)+-^^B.A_^ 

 \.X-l.X-2.X-3-p. {i+hy 



- iiq 



1.2.3.4 ^ A,-3 



\.X-l.\-2.X-3.X-4.\-5 „ (l+&)6 

 "^ 1.2.3.4.5.6 ^"X-5 ' 



+ &c., 



where the series is continued for so long as the factor in the de- 

 nominator is positive. It should be observed that this factor 

 really divides out, and that the rule just mentioned amounts to 

 this, viz. that when X is odd, the finite series on the right-hand 

 side is to be continued to its last term ; but when X is even, the 

 series is to be continued only to the last term but one. And 

 G;^ being thus defined, the expression for M. (see equation 164, 



in which I have written for A its value ^ — r) is 



1 + 



M,= (l-l-^.y->{G,+ (i + 2)i^G,,, +(i + 3) x^2(^yG,,. 



,. ,.i.i-\ /l + ^'V^ „ "1 



