264 REPORT — 1897. 



By Abel's theorem, when the two equations 



are simultaneously given, a and b are algebraically determined in terms of 

 the given quantities Xi,X2,X2,x^. 

 Now write 



^■^f{^,(x^) + ^,{x,)=v; ^■^n^x{x3) + ^,{x,)=v'. 

 Then from (1) it follows that 



^^ l^i{a} + ^,{b)=v + v'. 



If now in (2) we consider a;, and x^ as functions of it and v and write 

 Xi=X{u,v),X2=\i{u,v),a.nd similarly in (2') write Xj=X(u',v'), x^=\i{u',v'), 

 then it follows from (3) that a=\{u + u',v + v'), b^=\iiu + u',v + v'). 



Since a and 6 are algebraically expressible in x^,X2,x^,x^, it follows- 

 that \{u + u',v + v') and X.y{u + 2i',y + v') are algebraically expressible in 

 terms of \(u,v,) X^(u,v), \(^ii',v') and Xi(u',v'). 



The general theorem may be expressed as follows : 



■LetC ^^^=^.(x), i=(0, 1, . . . m-2), 

 Jo^/ X 



where X=/'(rc) is a rational integral function of the 2)/ilh or 2m— 1th 

 degree in ,r, then, if between the ?« — 1 quantities X(i,x^, . . . x„_„ and the 

 quantities ■tfoj^'u • • • '^ii-2 ^^e following equations exist simultaneously, 



(I.) ,(, = 4..(.^■o) + 4'.(•^•l)+ • • • +^.{^,n-^, (^=0, 1, . . . m-2) ; 

 and if 



(II.) .ri=Xi(wo,«i, • • •'''m-2). (i=0, 1, . . . m-2) ; 

 with equations similar to (I.) and (II.) Avith accented w's and a;'s, then 

 these functions enjoy the same property as do the trigonometric and 

 elliptic functions, viz. : 

 The functions 



X,(?<o.+V.^*l+«l'. • • • "m-2 + «'m-2) 



may be algebraically expressed through the functions 



'^i(»o.«i. • • • ■^'m-2) and A,(h'o,«',, . . . u'^.^), 



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



(24) Integrals of differmtial equations. — Euler's theorem sets forth the 

 complete algebraic integral of a differential equation of the first order 

 with two variables, which have been sej^arated in such a way that 



dx^ dx.2 



— =- "r =^ > 



\/ X 1 n/ X 2 



where X, and X, denote the same rational integral function of the fourth 

 degree in x, and x.^ respectively ; and Euler ^ showed that the algebraic 

 integral was an equation of the second degree between the two quantities 

 aj, +,r2 and .Tj.x.j. 



Abel's theorem sets forth algebraically m — \ complete integrals (inte- 

 grals which involve m — \ arbitrary constants) of m — 1 differential 



' Euler, InstitHtionea. Calc. Int. t. i. cap. vi. § 2. 



