406 MISCELLANEOUS PROPOSITIONS. 



Again, in the general expression (2) \eif(x) =F*(x), and 

 2 = 0; then 



and assuming that F n+1 (x) does not vanish between the 

 values a and a + k, we have 



In (2) put q = ; thus 



F n+1 (a + eh) 



Mfrnoires de I'AcadSmie... de Montpellier, Vol. 5, 1861. ..1863. 

 374. Expand V(l *) sm" 1 ^ in powers of x. 



Assume V(l *) sin~ : ic = ^ 



Differentiate both sides with respect to x ; thus 

 x sin" 1 a; 



- + " 1 '* + - 



that is 1- 



JL ~ Z/ 



= A 

 therefore 1 a;' x (A + A^x + -4^ + . . .) 



Equate the coefficients of of ; thus if r be greater than 2 

 we have 



therefore (r - 2) A r-1 =(r+l) A rH . 



Also we can see by expanding V(l a; 2 ) and sin" 1 a; and 

 forming their product that 



4 = 1, 4=0, 4=-l; 



