518 LAPLACE. 





therefore dx^dQ^ di V25 1 1 + — r— + -^ 



\ 3V25 65 



The limits of t corresponding to the limits and go of a; will 

 be — 00 and + 00 . Therefore 



/-«> /"» ( ^t ^ \ 



Jo J -00 I 8V2s 65 J 



By integration we obtain 



y. 



==As'^^e-'\^'27r \l+ Y^+ ...\ . 



Laplace says we may determine the value of the factor 



very simply thus. 



7? O 



Denote it by 1 ■] H -7 + ... so that 



•^ s s 



7/, = As'^'^e-W2^h + j + ~ + ...\ 

 Substitute this value in the equation 



thus 



' ly^^ _, r B C ] , B , 



therefore 



n.?+f+...){;-H)-('n)_^} 



8 S 



B B-2G 



--^ + — ^j— + ... 



And 



~ 126'^ "^12s^^ ••' 

 Thus 



f, B G \ ( 1 1 I ^ , B-20 



s ' ^^ ' J 12s' ' 12/ s' ' s 



3 



