256 REPORT— 1897. 



Hence upon integrating 



du. 



(HI.) 2«'.= 2 '^k,y,)dx, = \r{u) 



.=1 -=ij.T.o,2^.o J 



We note that the number fi does not depend upon the function R(a;,y). 



(14) We thus have the sum of fi integrals expressed as the integral of 

 a rational function of the parameter u. This integral depends upon the 

 nature of the function 'R{x,y) being, first, a constant ; secondly, the differ- 

 ential of a logarithmic function which is equivalent to a determinate 

 algebraic function ; thirdly, a logarithmic expression which likewise is deter- 

 minate, when the integral ^{x,y)dx are special integrals of the first, 



second, and third kinds respectively. 



The discussion of these special integrals, the normal integrals of the 

 first, second and third kinds, is found in Forsyth, ' Theory of Functions,' 

 p. 443 ; Harkness and Morley, * A Treatise on the Theory of Functions,' 

 p. 435 ; Neumann, ' Theorie der Abel'schen Integrale,' p. 245, &c. 



(15) Neither ' has proved that any algebraic curve may by means of 

 birational transformations be transformed into another curve in which 

 the highest singularities are double points with distinct tangents. We 

 may therefore assume that the curve •^{x,y)=^Q has no higher singularities 

 than these. Upon this hypothesis the most intricate integral that arises 

 may be expressed linearly in tei'ms of the normal integi'als of the first, 

 second and third kinds, with the addition, perhaps, of an algebraic 

 expression. 



Abel allowed the curve x(a.',7/)=0 to have any kind of singularity ; 

 and hence the expression for the algebraic and logarithmic functions that 

 stand on the right of formula (III.) in art. 13 are necessarily very com- 

 plicated. By making use of the methods mentioned above, this complexity 



is avoided, and the representation of the integral r(u)du may be obtained 



in comparatively simple form. 



(16) Denote the /> linearly independent normal integrals of the first 

 kind by v^{x,y){h=l, 2, . , , 2^) ', then, as in art. 13, 



^x.,y. __ 



dv,Jlx,y)dxz=Q (mod. const.), 



•=ijx\y' 



or VI dv,^(x,y)dx-\-2i^\ dVf,(x,y)dxr=S) {:moA. q.oxv?,\j.), 



where ^4- 2= A*- 



The points a;",, f, {k = 1, 2, ... 9) ; 33",+,, 2/V« (->• = 1, 2, . . . jo) and 

 the points a;,, y, (^'= 1, 2, . . . q) may be chosen at pleasure ; but the 

 remaining^; points a;,+^ 3/,+, (c = 1, 2, . . . ;:>) are no longer arbitrary, the 

 a;'s being the roots of an algebraic equation of the ^jth degree 



3?"+ a^x^-'^ -^ a^x^-'^ + . . . -I- Kp = 0, 



* Nother, Math. Ann. bd. ix. p. 17 ; see also Halphen, Bulletin de la Soc. Math, 

 de France, t. iv. Dec. 1875, and t. iii. Feb. 1875 ; Bertini, Mevista di Matem. 1891, 

 and Math. Ann. bd, xliv. p. 158 ; Poincare, Compt. Rend. July 1893. 



