328 REPORT— 1873. 



where 



C-E-D C+E-D_ C+E+D C-E+D 



F-1 ~ 1' F + 1 ~ =•' F + l ~ ^' F-1 ~ *• 



If we put , _ c^—y"^ 



1— aV 

 where a is a root of the equation 



l_2Ee'^ + ^'=0, 

 we are able to deduce 



v'(2(F-l))^^ 1^1^:217+?; = VRF + 1))^^ Vl-2E7/^+yV 



By this substitution Gopel remarks that we obtain an equation perfect in 

 Bymmetrical form with respect to the variables. And, lastly, putting 



_ / 1-y' Y . _ / I - v" Y 



he equations become 



dx \^1 — m.jc dx \/\ — m^x' 



V «1 - .r)(l -ma')(l - w^.r)) "^ V(^'(l - -rOCr^m^'Xi - »"ia;")) 



rf.rvl— m^a? (/a?'v 1 — ^H^a;' 



V (41 - .t')(l - rnx){l - m.^x)) ^ {x\l - x'){l - mx'){l - m.jv')) 



when 



E + 1 E, + l E,4-l 



"^ = 5^111' '"i = e;3T' "'^ = e:3i- 



Hence the solution of the hyiierelliptic differential equations of the first 

 order is easily obtained. 



Section 14. — In connexion with this part of the Report we may consider a 

 vejy beautiful method of integrating a certain system of hyperelliptic differ- 

 ential equations given by Jacobi in the 32nd volume of CreUe's Journal. 



Let 



Yx"~Y^x-'+Y.^x"-'. . . . ±Y„='Rf + 2Sy + T=0 



be an equation represented in two different ways, where Y, Y^ . . . . are, of 

 course, of the second, and E, S, T of the nth order in y and x respectively. 

 Then this equation, differentiated, manifestly gives 



^^-'^ , 2(7^ 



Ri/ + S ^ nYx''-^-{n-l)Y^x"-\ . . . +Y„_i ~ 



Let .r„. be one of the n roots of the algebraical equation ; then this 

 gives us 



y<„ , 2dy 



V'S^' - R„.T„, "^ Y{x,„ - xj(x„, -x.^)....(x„- X,:) ' 



