ox THE HISTORICAL DEVELOrMENT OF ABELIAN FUNCTIONS. 265 



equations of the first order with vi variables, in each of which the m 

 variables are separated. 



Taking the next simplest case, let \=/(x) denote a rational integral 

 function of the fifth or sixth degree. Then the two transcendental 

 equations (see the preceding article) 



<I>(.r 1 ) + ^x^) + ^X3)=^a) + ^b), 



owing to Abel's theorem, take the place of two algebraic equations 

 between the five qualities ,t,, Xo, .T;^, a and b. Consider a and b as 

 constants. Then, when we differentiate the two equations just written, 

 the terms involving a and b drop out, so that these quantities appear as 

 arbitrary constants in the transcendental equations, or in the algebraic 

 equations which take the place of the transcendental. 



Hence we have the following theorem : Lety(a;) be a rational integral 

 function of the fifth or sixth degree in x, and write y(.r,)=X(, (i=l, 2, 3), 

 then the differentia] equations of the first order with three variables, 



er.) -^^ + ^- + — :_•' = : J L + .--^^^ + ■^ * =0 



n/X, v/X2 v/Xj n/Xi v/X2 VX3 



have two complete algebraic integrals. 



This theorem is easily extended to m — 1 linear differential equations of 

 the first order with in variables, in each of which the variables are 

 separated : 



/^ \ x\dx^ .:^2yx.2 x^dx, ^ q 



(i=0, 1, 2 . . . m-2), 



where X,, Xj, . . . X,„ denote the same integral functions in x,,a:.2, 

 . . . a*,,,. Jacobi closes the ' Considerationes generales,' &c. with the 

 remarks : ' We know that Lagrange, starting with the differential equa- 

 tion between two variables [art. 7] came to its complete algebraic integral 

 through direct methods of integration, and so by a new and singular 

 method demonstrated Euler's theorem ; and so we think it worth the 

 while to investigate through direct methods of. integration the two 

 complete algebraic integrals of the system (rr.) above, or more generally 

 the m— 1 complete algebraic integrals of the system (2.), and thus adorn 

 Abel's theorem with a new and no less singular demonstration.' 



At the end of the ' Note von der geodatischen Linie auf einem 

 Ellipsoid,' &.C. ('Werke,'bd. ii. p. 59), Jacobi finds that by making use of a 

 certain substitution he was able to extend the remarkable relation dis- 

 covered by Legendre between the complete integrals of the first and 

 second kind of two elliptic integrals whose moduli are complements to 

 each other to all hyperelliptic integrals ; and this same substitution, he 

 says, led him to the Abelian theorem itself in a way and through con- 

 siderations which are absolutely different from that of Abel. These 

 considerations originate from a mechanical problem. 



The elliptic movement of a planet, or even the motion of a point in a 

 straight line, may be expressed through an equation between two elliptic 

 integrals. Wc have two methods of treating the same problem, of which 

 the one represents the solution in a transcendental, the other in an 



