ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 327 



sdp ■\ — dq sdjj — —dq 



^=^'^' ^=^^" (^) 



Also, using the same notation, the last four equations of section 11 may 

 be written thus : — 



(l-2Ep^+p<)s^-2(r(l+^Y)-CCpHg") + 2Dpg)s- 



+ {l-2Ecf+q*)=0, • (2) 



2VE^^.(6;//=(l-2Ep^+pV-(l-2E2H2^)-„ . ... (3) 



f=(b+bxi+rq')-(t(F-+i')+2ic-c,)M, (4) 



;/.^=(6-6J(l+^Y)-a(^H2^)+2(c + cJ^jg, (5) 



•where E, F, a, b, &c. are constants, whose values wiU be found in p. 299 ; 

 hence, by addition, we find 



Fa+2> Y)-C(p' + q ' )+2J)pq+ s/W^l . ,p^ 

 l-2Ep^+p^ 

 l_ 'P(l+j)Y)-C(}>' + q')+2J)jyq- Vl^^IT. <p ^{, 

 s' l-2Ef + 5' 



Moreover equations (1) may be written 



{'''^ + T) (dp , dq\ {'^P-t)( dp dq_\, 



^ \2\p'^ 2£i.q)'^ ;// \2A« 2A2/ 



Putting here _2/Ar+zAy _'>/^z—zA>/ 



^ 1-2/V"' ^~T-2/V' ' 

 where also 



Ay = V l-Ey' + y\ Ar = V T^ Ez^~+z\ 



and remembering that 



& 



dp di/ dz dq _ dy dz 



Ap ~ Ay Ar' Aq ~" Ay Ar' 



we sejiarate the variables, and obtain 



^^ ^^ ''V(l-2E/ + 2/^) V Vl-2E,/+W 



-t- v'|2rr+i)i '^~ .A A-^?-5?!±=' = s/h^dy, 



^^^ *^ ^ -"VCl-SErH-') V l-2E/+r'' 



