Applications of Khdh Theorem, 119 



1 — C* = l*! 3^2 



1 + C' = — JT, — J*2 



2 = J?i J72 — JTi — JTj 3 = (1 — ^i) (1 — ^i) 



2 + ^2 



When Xc^ = \ x^ = — infinity 



^2 = Y ^1 = -5 

 - 1 _ *? 



/ dx _ ^ 1 .3a:7 1 . 3 . 5 ot^" 



v^i::::^ ""^"^ 2.4 "^ 2.4.7 "^ 2.4.6.10 ■^'**^- 



Making a: = — this series gives \I/ a: = -508264 



or = -— ij/a; = -334901. 



Making x ■=. — y 



P dx __ p dy 



Now, let T-—-^. = u 



1+y 



dy 



y v^ny -^j" +3.7" ^3.6.13" +'^''-/ 



If \I/' — denote the integral 



y^ dy 



taken from ^ = to ^ = 00, the preceding series gives when 

 :r=-5, w=^, ^|,a= -893917 -^^''^ 



^="-T'''=Jl''''^= 1-06728 — vl/ -^. 



By the theorem of Abel, if ^j and 0*2 are connected by 

 the equation 



x^ = 1. (See line 4 above.) 



«i4'^i + «2^2 = constant, or 

 n'^dx p'^dx 



