326 REPORT— 1873. 



and then it is seen without much difficulty that 



^2r«H2r'«-.^p,^^g, - S'cZP') =«,T, + h;i\ + c,\\ + (7,W„ • 

 where Tj, U„ V,, W, are the values of 



2r(»+(j)+i)K+(G+4)L)=' , „ . , . „ , 



Se when 6 is even and tj is eyen, even and 



»/ odd, &c. 



Hence we find 



P'rfS'-S'(?P'=«PS + 6F"S"' + cQIl4-<?Q"'Il"'+a,P'S-|-i,r"S" 



■where such quantities as P'Q are excluded according to the law given in 

 page 290, and a, h, &c. are of the form fdu-\-f'(lu', where / and /' are con- 

 stants. But since Q'R', Q"R" can he expressed in terms of P'S' and P"kS", 

 and also QR, a"'Il"' in terms of PS, P"'8"', we shall have 



P'cZS'-S'(?P'=aPS + 6P"'S'" + fl,P,S,-|-&^P"S". 



Putting it-j-K+L for u, we have 



P'tZS' - S'tZP' =aPS + &P"'S"' - a^P'S' - 6,P"S" ; 



whence 



PWS'-SV?P' = «PS + 6P"'S"' ; 



and changing n into 2<-}-A + B, 



P' JS " - S"(/P" = «P"S" + 6PS, 



the coefficients are easily determined ; and we have, finally, 



P'(?S' - S'c?P' =e^^PS + }^^ P"'S"', 

 p ic k p 



Y'dS" - S'VZP" =P'y^'|,'p"'S"' + ~% PS. 

 (d"'/i.' k"'p"' 



Section 13. — Wo shall now show that from these equations the hyper- 

 elliptic differential equations can be deduced. We shaU give the outline of 

 the calculation, referring the reader for the details to the original memoir. 

 Prom the equations last given, we have 



(P'dS' - S'fZP') + (P"tZS" - ii"dF") _ Fdp'+p"dl'' _ 



p"'S"'+PS ~ k"y' ^'*' 



(P'<7S' - S'rtP') - (P"<ZS" - S'V?P") ^ lc'dp'-p"dl-" _ 



T)"'C" T)0 7'" '" *'* 



Putting 



o 



P"'S"'-PS k"p 



S' _ S" _ F _ 



F"S"'+PS _ F"S"'-PS _ 



P'P'f 9' PT' 



the last equations are transformed into the following 



