1880.] Mr. R. C. Rowe. Memoir on AbeTs Theorem. 519 



obtained there may be found for the sum of any number of such 

 functions each multiplied by any rational number, positive or nega- 

 tive, integral or fractional ; secondly, an investigation of the con- 

 ditions that the algebraic expression obtained for the sum of the 

 integrals considered in Section I may reduce to a constant ; and lastly, 

 a discussion, as an example of all the results of the paper, of the 

 case 



where = i-L^z ? X and O being rational integral functions, while 

 3 (x) 



y is a root of the equation 



x =y n -f=0, 

 and e= qn _ iy n-l. + qn _ 2yn -z + . m . + qi y + q Q . 



A considerable simplification is introduced into the second part of 

 this section by placing it after (instead of, with Abel, before) Sec- 

 tion II. 



In the Appendix will be found notes on three points in the paper ; 

 and a list — it is hoped complete — of the errata in Abel's memoir. 



There follows an addition from Professor Cayley, wherein it is 

 shown that the foregoing expression for the least value of the 

 number of conditions is equal to the deficiency (Geschlecht) of the 

 curve represented by the equation x(®, 2/) = 0- That this least value 

 is equal to the deficiency is a leading result in Riemann's theory of the 

 Abelian integrals ; the before-mentioned assumptions as to the form 

 of the roots of the equation y)=0 considered as an equation for 

 the determination of y are equivalent to the assumption that the curve 

 x(%, y) = has certain singularities ; and it is in the addition shown that 

 the resulting value of the deficiency, as calculated by the formulas in 

 Professor Cayley's paper " On the Higher Singularities of a Plane 

 Curve," Quart. Math. Journ., vol. vii (1866), pp. 212-222, has in fact 

 the foregoing value. 



