LAGRANGE'S AND LAPLACE'S THEOREMS. 119 



fx\ n 

 obviously f -J : whereas the first term of the expansion with 



/2\ n /aA~* 



the lower sign would be ( - 1 , that is [ -I . 

 W \2/ 



Now y- _--. , 



*V - 2 ~ 



, n 

 ~ 



1.2.3 " 



Let a; 8 = 4i ; thus we obtain 



1.2.3 



Change the sign of n; thus we obtain the expansion in 



, . . (l-V(l-4i)}-" ,, , 



powers of of | -- _ -| , that is of 



that is of 



Hence 



n (ra - 4) (n - 5) ^ 

 1.2.3 



Hitherto we have put no restriction on the value of n; 

 but let us now suppose that n is a positive integer. 



If we expand {1 + V(l - 4*)}" and {l-V(l-4i)} n by the 

 Binomial Theorem, we see that the sum of the two expressions 



will be a rational function of t which will be of the degree - 



m 



n 1 

 if n be even, and of the degree - if n be odd. 



