280 REPORT— 1897. 



formation, multiplication, and division that are found in Jacobi's 'Fund. 

 Nova.' Among others we may mention a direct method of the trans- 

 formation of elliptic functions of the third kind, without presupposing, as 

 is done by Jacobi, the formula of the transformation of functions of the 

 second kind. 



(41) Jacobi, in the eleventh section of his memoir ' De functionibus 

 duarum variabilium quadriipliciter periodicis,' states without demonstra- 

 tion that if we have 



X = \{n, v), y = \{u,v\ ■ ■ 



then the functions 



are by Abel's theorem obtained as the roots of a quadratic equation 

 Ua;,.2 + U'a;„ + U" = 0, 



in which U, U', U" are rational functions of .r, y, \/X,\/Y (see art. 37) ; 

 and from this it is seen, vice versA, that x and y may be obtained from 

 a;,, and y,, by the solution of algebraic equations. 



Hermite (Crelle, bd. xxxii. pp. 176 and 277) finds that the coefficients 

 U, XT', and U" are of the form P + Q ^(a;)Ay, where P and Q are rational 

 functions of x and y, and where 



A(a;) = s/ X (above) = s/x{\-x)(\- k'x) ( 1 - X'x) ( 1 - fj^^x). 

 Let i^>J — 1, i^, ^3%/ — 1, ^4 be the four periods of the integral 



f («+/3a:) 



A(x) 



-dx, and i\\/ — 1, i'.^, i'sV — 1, V \ the corresponding periods of 



-> " ( ^' dx : then the simultaneous roots of tlie two equations 

 J A(x) 



(A.) Ua:,.2 + TJ'a:„ + U"=0, U^„2 + UV„ + U"=0 



are expre.ssed by Hermite through the formulae 



3;=,\ M + 



mi^s/ — 1 + m'i.2+'m"i3s/ —1 +7?i"'ii, 



li. 



V+-- 



mi\\/ — 1 +7n'i',^+'m"i\\/ — 



\+m"' i'\ 



'=Ai(' 



n 

 mi^ \/ — 1 +mi2 + m"i3\/ — 1 +7n"'i„ 



v + 



u + 



n 



rni' , \/ — 1 -f m'i' 2 -f m''i'^ \/ — 



l+m"V \ 



where m, m', m", and m'" may take any of the values 0, 1, . . . n — \. 

 For brevity Hermite writes 



I=mi', v'— 1 ■\-m'i^ + m"i.i\/ — 1 +m"'i^, 

 and I'=:?ui',\/ — 1 +m'i' .2 + '»i"i' i^ — 1 +OT"'i'^, 



and by/(a;,3/) is denoted any rational symmetric function of x and y, 

 B.Txd p, q, r, s are used to indicate any four of the roots a;"^!. 



