1883.] 



x \(ic-x 2 ) 



of Elliptic Functions, 

 dD 



4>Dx+(ic-x 2 ) 



dx 



-{l-J-icf 



+2rJ-ic(l-ic) 

 dD 



2D-, 



19 



dD 



dx 



(ic — Sx 2 ) D — (ic — x 2 ) x 



dx 



Now D = (i - 1 + J 2c) {(ic - x 2 j 2 - 2V J- ic (1 - ic)} 



+ (i-l - J2e) (ic - x 2 ) (1 - J ~^ic~) 2 x, 



dD r _ . 



fa = (» -1 + V2c) {- 4 (ic - a- 2 ) a? - 4ay - ic (1- ic)} 



+ (i - 1 - J 2c) (ic - Sx') (1 - J^icf; 

 therefore 



Wx + (ic - x 2 ) -j- = (i-l + sj2c)J—ic (1 - ic) (-4») (ic + x 2 ) 



d 



+ (i-l- J 2c) (ic - x 2 ) (1 - J- ic) 2 ,(ic + x 2 ), 



dD 



2D - x ^- = 2 (i - 1 + 72c) O'c - # 2 ) (tc + x 2 ) 



+ (i — 1 — 72c) (1 — V — i'c) 2 a? (ic + x 2 ), 



dD 



(ic - 3x 2 ) D - (ic — x 2 ) x ■^— 



therefore 



= (i-l+ J2c) (ic + x 2 ) {(ic - xj + 2a; 2 J- ic (1 - ic)} ; 



dy i — l + c + ic,. , ... 

 Ix = D> {%G + X) 



x {(ic - a; 2 ) [(t - 1 + J2c) J^Tc (1 - ic) (- «c) 



+ (t-l-72c) (ic-a; 2 ) (1 -V^) 2 ] 



4- 2xJ^Tc (1 - ic) [2 (i -1 "■+ ^27) (ic - a; 2 ) 



+ (i-l- N /2"c)(l-V^ic) 2 a ; ] 



- (1 - J-ic) 2 (i - 1 + V2c") [(ic - a; 2 ) 2 



+2ay r ic~(l-ic)]} 



" * C (ic + a; 2 ) {(ic - x 2 ) 2 (1 - J~m) 2 (- 2 jtc) 



D 2 



i — l+c + ic , , 



p (ic + x 2 ) (1 - J- ic) 2 (- 2V2c) {{ic - x 2 ) 



+ 2x 2 J- ic (1 - ic) (l-sj- ic) 2 (- 2 J 2c)} 



i.^)(l-J^ic) 2 (-i 



+ 2x 2 sJ^7c~(l-ic)} 



2—2 



