LAPLACE. 477 



also diminishes as a increases. We will demonstrate tliis. 

 Put r for w — 2m, and denote the fraction by u ; then 



idu^ (1 + cr' + (1 - ay (1 + ay-' + (1 - cy-' 



uda '' {l+OLy-{l-ay ^ (l + a)"-(l-a)" * 

 Thus 



where 2; = :j . We have to shew that this expression is nega- 



tive : this we shall do by shewino^ that -~. — :i — "^ increases as 



successive integral values are ascribed to r. We have 



(r + 1) (3^+1) r{z'-'+l) 

 z'^' - 1 z'-\ 



_ (r + 1) (/' -l)-r (z^' - 1) (z^-' + 1) ^ 

 {z""^' - 1) {z' - 1) ' 



thus we must shew that z^"" — 1 is greater than r {z'^^^ — z""^). 



Expand by the exponential theorem ; then we find we have to 

 shew that 



{2ry is gi-eater than r | (r + 1)^ - (r - 1)^ I , 



where ^ is any positive integer ; that is, we must shew that 

 2i>-i ,J'-i is greater than ^r"'' 4- i^ (/^ ~ ^H/> - ^) ^,^-3 _^ ^^^ 



But this is obvious, for r is supposed greater than unity, and 

 the two members would be equal if all the exponents of r on the 

 right hand side of the inequality were^ — 1. 



We observe that r must be supposed not less than 2 ; if r = 1 

 we have s'^'" — 1 = 7- (s*^^ — s*^^). 



We have assumed that r and n are integers, and this limitation 

 is necessary. For return to the expression 



(1 + a)' - (1 - a)' 



(1 + a)" -{\-a) 



n J 



