258 REPORT— 1897. 



From these relations it is seen that as soon as a of the x's are given, 

 the remaining fi — a may be determined in terms of the known ones. Abel 

 showed how to effect this determination, and in general that f^ — a = ■y. 



In two special cases considered by Abel this number is less than y. 

 (See also Rowe, Memoir on Abel's Theorem, 'Phil. Trans.' 1881, p. 731.) 

 Professor Cayley, in the ' Addition to Mr. Rowe's Memoir,' proved that y 

 was always equal to the deficiency of the curve x(^)2/) = 0) whatever its 

 singularities. 



Professor H. F. Baker has recently proved the same theorem by means 

 of graphic methods in the ' Cambr. Trans.' xv. Part IV. ; see also 

 'Math. Ann.,' 45, p. 133. 



As a special case Abel gave the equation xi^'V) = the form 

 y» + Po = 0. 



The form of the integrals whose sum is to be expressed as in 

 formula (I.) is 



I 



Mx)clx^ 



For the hyperelliptic functions {n = 2), when Pq is of the 2?n — P' or 

 2-in}^ degree, Abel showed that ^ — a = m — 1 . 



(19) Mathematicians were much interested in the new functions which 

 must be introduced in connection with the Abelian integrals. The 

 Academy at Copenhagen wished to see these functions extended to all 

 integrals of algebraic functions, which are included in Abel's theorem ; ' 

 and in regard to this wish Jiirgensen, Broch, Minding, Rosenhain wrote 

 some very important memoirs. The value of these memoirs, however, on 

 account of their less generality was much diminished when Abel's great 

 paper was finally published in 1841. 



Minding, in two short papers (Crelle, bd. ix. p. 295, 1833, and bd. xi. 

 p. 233, 1834), showed how to represent the algebraic and logarithmic 

 functions of Abel's theorem for the special cases in which the algebraic 

 functions satisfy an equation of the third degree. 



Jiirgensen (' Sur la Sommation des Transcendantea a difierentielles 

 algebriques,' Crelle, bd. xix. p. 113), took, as the subject of integration, 

 the quotient of two functions T(x, Sj) and Q{x, s,), where P and Q are 

 integral functions, and where z^ is a root of an equation that is similar to 

 Abel's x(^>2/) = 0. 



After reducing ^ }' ' ~'( to a form -^'-jj where \ and v denote in - 

 Q {x, 2.) r{x) 



legral functions (see Liouville, ' Note sur la Determination des integrals 

 dont la valeur est algebrique,' Crelle, bd. x. p. 347),^ he considered a 



sum of integrals of the form — ^ ' t ' where the summation is taken over 



the fi roots of the resultant of two algebraic equations. This sum he 

 expressed in the form of an algebraic and logarithmic function. 



In a second paper (Crelle, bd. xxiii. p. 126) Jiirgensen denoted by X(x, yj 



' Cf. Jacobi, Gesam. Werke, bd. ii. p. 517. He does not mention Jiirgensen. 

 ^ See also references cited in art. 11, and a paper by Liouville, Sur Vintegratioii 

 d'une classe de fo notions transcendaiites, bd.xiii. p. 93. 



