( 440 ) 

 ul)()ve givcji \aliies of d, <l' , etc. (expressed iii ./■ and y) : 



\^l — "lw-\-'ly 1 — x—yj m 



If we now pnt ,/•-!-ƒ/== y> and .c — .y = 7, <<^ simplify tlie further 

 ealcnlation, we get for the logarithm: 

 ^ /l+2pl4-^ 



2 o 4 o () / 



111111 



+ Pi-,l'' +-^P' +^P' -^--r^P' -^-^P' +^-p' +-I 



+ 2^+ ^W)-f 3(8^4- ^(1«7') + J (32^^) + ^(64g«) + ^(128^^)+ ...| 

 11,1 1 1 1 



i. e. 



= 3^>-^ .:V+3 -V— 4 -15?'+ 5 -^Sp'^— ^-.6:V + ^ • 129;,' 



+ 3.;+ ^ . 'V+ 3 • V+ 4 • 1^7^ r [ . 33y^+ ^ . 63<y» ^ J . 129 ./ 

 or 



=r6 



2 2 2 2 4 2 



lip' +9» 7 p'—q" 43 p' +9' 

 "^ "5 2 2 ^2 ^ T ~^ 



Hence w^e have after re-substitntion of .t' ~\- y and x — y for p and ^ : 



.. - 1 (2^^) + (,7-H 3/»v/^) — 7 (4^'// + 4^y«) + 

 2 4 



11 7 43 , x{l—v) 



5 2 / m 



in which the expansions have not been carried further than is 

 necessar}^ for the determination of the terms with t'. After division 

 of both the members by x we ^et : 



/I \ 43 1— V 



(l-y) + (.7.^ + 32/^) — 5(..V+^) + ll 7'''' + 2<v^ -21..V + -7-'^''' = 



5 ' y '7 m 



Substituting into this the expressions with t for ,v and ?/, we get 

 after subtractiim of 1 — y from both members: 



