524< LAPLACE. 



rical maximum value of ( —. — ~] to the first is less than f J) . 

 Similarly the ratio of the third numerical maximum value to the 

 first is less than f^j . And so on. 



Next suppose that we require the coefficient of a in the 

 expansion of 



[4 + -4i + -^2 +... + - + 1 + « + •••+«""'+«""' + ^4'- 

 [a a a a ) 



The coefficient of a'" in this expansion will be the same as the 

 coefficient of a"*" ; denote the coefficient of a"" by A^.. Pat e^^ 

 for a and suppose the expression to be arranged according to 

 cosines of the multiples of 6 ; then 2A^ cos r6 will be the term 

 corresponding to A^ (a*" + a"*"). If we multiply the expression by 

 cos W, and integrate between the limits and tt, all the terms 

 will vanish except that for which r is equal to I; so that the 



integral reduces to 2Ai j cos^ Wd6. Hence 



cos WdO. 



1 



We put; as before, m — 2n + 1, and (f> = ^0; thus we have 



As before assume 



/sin m4i\^ 



c 



sin md>Y „ _,, 



sm 



V 



then </) = .. , \ — TyT , approximately. 



Hence the intoOTal becomes 



to' 



2 "^'^'5 f^-,^^^ 2?^y6 ^, 



TT 



Vis K- 1)1 J" ^"V(«K-i)} 



