LAPLACE. 525 



As before we take and oo for the limits of t, and thus 

 neglect all that part of the integral with respect to which is not 



TT 



included between the limits and — . Hence by Art. 958 we 



m 



have finally 



2 77ZV6 Vtt -,-J^ (2/z + l)V3 -^-^, 



;; sj[s {m' - 1)} 2 ^ ' ""^ ^/[n {n + 1) 2s7r} 



Suppose now that we require the sum of the coefficients, from 

 that of a-^ to that of o} both inclusive ; we must find the sum of 



2^j+2^z.i+2^i_2+... + 2^^ + ^,: 



this is best effected by the aid of Euler's Theorem ; see Art. 384. 

 We have approximately 



ri 11 



r? 1 1 



therefore ^^ u^ = I u^dx + « ^^a; + « ^o > 



therefore 2^^^^ Ug; — u^ = 2 I ujix + u^, 



J 



Hence the required result is 



We may observe that Laplace demonstrates Euler's Tlieorem 

 in the manner which is now usual in elementary works, that is by 

 the aid of the Calculus of Operations. 



966. Laplace gives on his page 158 the formula 





00 



/. 



1 



x^ ^ e^ dx 







He demonstrates this in his own way ; it is sufficient to obseiTe 

 that it may be obtained by putting x for sx in the integral in the 

 numerator of the left-hand side. 



