LAPLACE. 519 



Hence, equating coefficients, 



^ ~ 12 ' 288 ' 



The value of A in the expression for y^ must be determined 

 by some particular value of ?/g. Suppose that when s = yx we 

 have 2/^= Y. 



Then Y^A\ x^e'^dx^ 



J a 



thus A = 







Y 



/•OO 



Hence 



Vs- p i^+ 12s'^288/"^*"'i* 



I x^ e ax 



^ 



The original equation can be very easily integrated; and we 



obtain 



y,^ rOL6 + l)0+2) ...5. 



Hence, by equating the two values of y^, 

 (/i + 1) (/I +2) ...s = 



•^ 



a;'^ e"'" cfa? 



It will be observed that 5 — /z is assumed to be a positive 

 integer, but there is nothing to require that 5 itself should be an 



integer. 



963. One remark must be made on the process which we have 

 just given. Let (s) denote 



*■ "T T(r>_'ooo^2'" 



will be denoted by <f>{—s). 



Now Laplace does not shew that 



