282 REPORT— 1897. 



In a letter to Liouville (loc. cit. p. 361), Hermite states that the 

 representation of these functions is attended with great difBculties. 



By supposing successively /(a;, 2/)=a^+y and /(x, y)=x. y, the pre- 

 ceding theorem will give, expressed by a sum of nth roots, in number 

 n* — l, the coefficients of an equation of the second degree, whose roots 

 will determine those of the proposed equations. These n^—l roots will 

 be expressed rationally in terms of four of them. 



Hermite next discusses the division of the periods. 



In the second paper mentioned above, Hermite derives Jirst certain 

 theorems, from which he deduces, among others, Jacobi's formula for the 



algebraic expression of sin am (x) by sin am i:^], and Abel's fundamental 



properties of elliptic functions which relate to the addition of the argu- 

 ments ; other formulae, which involve Jacobi's H and 6 functions are 

 given. 



Applications are then made to functions of two arguments and four 

 periods. Hermite writes the integrals of the third kind in the form 



(I-) 



\\x — a X — bj Ax \y — a y—b/Ayj' 



the integral being subjected to vanish, when x=^0 and y=^0. Ax repre- 

 sents the square root of the Tpo\jnomial 2h^ +3h^'^ + 2h^^ + 2h'^'^ +2^&^- 

 After 



x=\(u,v) 2/=Xi(m,u) 



a=X(a,/3) b=X,{a,(5) 

 are substituted in (I.), this integral is denoted by IT (n,t\u,(i). When the 

 variables u and v are introduced into the integrals of the second kind 



(a?dx yHy\ , Ux'dx tfdy\ 

 {A{x)^A{yjJ ][A{xyA{y))' 



they are first denoted by {u,v\ and (ujv).^ respectively. 

 Two new integrals are defined by the relations : 



^i{u,v)=2pi{u,v\ + 3p^{u,v)i and E^ {u,v)=2Ji,{u,v)^. 



The following theorem is then derived 



U{u,v,a,l3) — 'n.{a,l3,u,v)=pg(uv — (3u) + uEJ^ii,,v) 



a formula in which is seen the law of interchange of parameter and 

 argument (art. 33). 



Hermite further defines a function $ by the relation 



*(w,'y,a,/3) = n(M,T,a,/3)-t-MZi(o,/3)-l-tiZa(a,/3)— c(ai;— /3m), 



where Zj and Z^ are certain functions of E^ and Ej respectively, and 

 c is a constant. 



Interesting addition-formulae are derived for the two functions 11 and 

 $ {cf. report made by Lame and Liouville, ' Comp. Rend.' xvii. and 

 ' Liouv. Journ.' t. viii. p. 502). 



(43) Tlte introduction of the theta-function. — The new functions of four 

 simultaneous periods which Jacobi had discovered were received with 

 great enthusiasm by mathematicians. The Academy of Sciences at 

 Copenhagen wished to see presented the analogous functions that are 

 connected with the integrals of all algebraic functions, to which Abel's 



