266 REPORT — 1897. 



algebraic form. We thus derive a new method of finding the fundamental 

 theorem for the addition of elliptic integrals. Jacobi says that by 

 generalising this method through the introduction of any number of 

 variables he obtained the general addition-theorem in a new and ready 

 manner ; and at the same time there was opened a simpler way, through 

 the application of suitable multiplications, of coming directly to the 

 algebraic integrals of the systems of differential equations (rr.) and (S.) 

 above. See also a paper by Haedenkamp (Crelle, bd. xxii. p. 184). 



(25) Following Jacobi's suggestion of the preceding article, Richelot 

 (Crelle, bd. xxiii. p. 354) extended Lagrange's methods (art. 7) and in a 

 direct manner integrated the differential equations (a.) and (2.)' above. 



By his methods two algebraic integrals ai'e found for Euler's differen- 

 tial equation, of which it is easy to prove that the one is a function of 

 the other ; while the two algebraic solutions of (cr.) are independent. He 

 found two similar solutions for the system (2'.) The general solution of 

 this system of equations he derived in the following form : In system 

 (2'.) let X=/(a;)=Ao4-Aia:-f Aa^;^-!- . . . Aana;-", and for brevity write 



F{x)={x — Xi)(x—X2) . . . (a; — a;„) and F'(x)=—^^''; if, next, the roots of 

 the equation. 



{ao + aiX + a,x-'-+ . . . +a„x''y—b'-{Ao + AiX+ . . . -f A2„a;2»)=0 



be denoted by x^, X2, . . . x,„ nii, 57^2, . . . 7n,„ and if the coefficients 

 aQ,aj, . . . b he determined through the Jirst n+1 of these roots, then 

 the n — 1 equations of condition, which must exist in order that the ?i — 1 

 remaining quantities m2,m^ . . . m„ be the roots of this equation, are the 

 n — l complete algebraic integrals of the system (S'.).''^ 



Richelot assumed that /{x) has, as factors, x — oq, x — a^, . . . 

 X — a„_j, and for brevity he wrote 



P (x-) = {x — m.j) (x — ?«3) . . . (x — m„) ; m^ = a^. 



' Change vi into « in the system (2.) in order to liave the system adopted by 

 Richelot, which denote by (2'.). 



' It is interesting to note here certain difficulties that the older mathematicians 

 experienced. Euler {Fnst. Cal. Int. t. i., cap. vi. § 2, prob. 82, scholion 1) says it is 



quite clear that transcendents of the formj — ' 



] VX+BX+ Cx' + Dx* + ¥jX* + Fa;* + Gx* 

 cannot be treated in the manner of circular and elliptic integrals ; for if the coefficients 

 are restricted so that the root may be extracted, the formula becoming 



f dx 



I '■ , it can in no wise happen that several functions of this kind be 



}a + bx + ex'- + dx' 



algebraically compared among one another. Lagrange tried in vain to extend this 



theorem. This paradoxical thought is easily explained : for two algebraic equations 



between the arguments of the integrals always satisfy the two transcendental relations 



between these integrals when the function under the radical sign is of the fifth or 



sixth degree (art. 21). These algebraic equations exist owing to the fact that the 



numerator of the integral which is of the first degree has coefficients which are 



always of such a nature that two of the arguments become roots of a quadratic 



equation, whose roots involve the other arguments algebraically. As often, therefore, 



as the transcendents unite within themselves a logarithmic and a trigonometric 



part, it happens through the algebraic equations that both the trigonometric and the 



logarithmic parts vanish independently in the two transcendental equations, so 



that one relation is given among the logarithms and another among the arcs. Of. 



Richelot (Crelle, bd. ii. p. 181). 



