LAPLACE. 595 



Let us consider first the integration with respect to z^ ; we have 



= (a^+a%') (z ^2'^i^2 Y , 2 2 <^xX' 



2 2 2 



a. a„ ^. 



a^ + a^ ^1 



where ^ = 2;, 2"^— ^~2^~2 • 



1 a/ + a^b^ 



The limits of ^ will be — x and 00 ; integrate with respect to 

 t : thus we remove z^ entirely, and obtain the factor 



and instead of the first two terms in w^ we have the single term 



ttj ci^ z^ 



We integrate next with respect to z^ ; thus we shall remove 

 ^2 entirely, and introduce the factor 



and instead of the first three terms in ic^ we shall have the single 

 term 



2 2 2„ 2 /» /v 2 2 ■N -1 



"'1 ^'2 ^3 ^3 j ^1 S I ^ 2X 2I 



< + <^"' l< + <^l' "^ ^3 ^2 I • 



Thus we have now on the whole the factor 

 where -rs = — 2 + -^ + ^ ^ ' 



- «3 «2 «i 



2 ? 



and the first three terms in u^ are replaced by the single term \\^. 



38—2 



