ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 271 



hyperelliptic integrals of the first order may be reduced into canonical 

 forms by means of certain substitutions, even when the factors of the 

 function under the square-root sign in the denominator of the integrand are 

 not all real. Jacobi also avoids the imaginary arguments introduced by 

 Richelot through the application of the substitution 2y=:\/ a + hz + cz'^-jr 

 is/ a — bz + cz"^, and which Richelot again reduced to real arguments by 

 means of Abel's theorem. When the degree of the function under the 

 root sign is greater than the sixth, and when this function contains 

 imaginary factors, Jacobi asserts that no one has found the substitutions 

 by which the reduction may be performed. At the close of this article 

 Jacobi discusses Euler's addition-theorem from a more general standpoint 

 than that taken by Lagrange (art. 7). 



(32) Addition-theorems for hyperelliptic integrals. — Jacobi (Crelle, 

 bd. XXX. p. 121) derived an interesting form of the addition-theorem for 

 hyperelliptic integrals of the second and third kinds. Let E. be a given 

 integral function of the 2nth degree in x 



R=aia;2''-f«2a;-""'+ • • • +1, 



and V an integral function of the 7ith degree with unity as coefficient of 

 highest power of x ; further, let a be a constant and 



(1) xV^ + n'^'R=(x — x^){x—X2) . . . {x—X2„+i) 



then it may be proved that ^ 



(I) '^^^[J^dx, ^ 

 -JJVa;,R(x,) 



=0, 



where w takes any of the values 0, 1, . . . to — 1, and where the upper and 

 lower limits of the integrals are two systems of roots of two equations of 

 the form (1), in which a and the coefficients of V have different values, 

 while R remains the same. 



Further, we may deduce the following expressions 



«=2it + l/. „n + «^„ 



(IL) 2 ^L^=A., 

 -^^ J\/a;iR(a;.) 



1> "2) 



where A, (*:=0, 1, . . . «-fl) are algebraic functions of a, a 

 «2n+i- Jacobi gives a rule by which those functions are easily determined. 

 Jacobi makes the theorem above more general by the introduction of 

 a new variable x^n^^, and the formation of new formulae 



i=2ri+2,. ^, ii + «^^ 



tz J V.-CiR(a;i) 



where the quantities a, a,, . . . ajn+i, ttan+a ^'"^ i^ow determined through 

 an equation corresponding to (1) 



(x-x,)(x-x.^ . . . (x-x.,„^,)=x'"'*^ + a^x-''^' + a.,x'"'+ . . . +a,,„,. 



For the integrals of the third kind Jacobi proves the following 

 theorem : — 



By means of n + 2 given quantities x^, x^, . . . .t„+i, «, let three 



• See Abel, (Evvres, t. i. p. 444. 



