124 Mr. A. Cayley on an Analytical Theorem relating to 



which agrees ; and when i is even, then for the power 2x — i 

 there are two|equal and opposite terms which destroy each other, 

 and the whole term in 0j is, as it ought to be, zero. For any 

 odd power 3.r — 1 (a;> 1) (inchiding, when i is odd, the power 

 2x — \ = i), the term vanishes as containing an evanescent factor. 

 And the expression for ©; is thus shown to be true. 

 1 write for shortness, 



where 



X=l+0(l+i), 



Y= -0(1 + A). 



Forming the expression for «(1 + ^)M,_!, 



= 0i + y J ©i+i + ^-Y^W ®'^^^ ^^-' 

 this is 



and we have 



X=(l + 0)(l + 6)-i, and .-. 



(x(n- f)) =(^)'((H-0){(H-0)(H-i)-J} /, 



Y=-0(l + i), 



We see that the expression for ( Xf 1 + y j j is deduced from 

 that of(Y[l + -r)) by writing therein 1 + in the place of ; 

 we have therefore 



. (x(i.f)y=(i.A)(Y(i4)y, 



