260 REPORT— 1897. 



The contents of the second head are indicated by its title ' De numero 

 minimo integralium ad quae numerus datus eiusmodi integralium reduci 

 potest.' 



In the third head he makes application of the preceding theorems to 

 the equation : 2^oy"+l^n=^- (See also art. 19.) 



Ramus (Crelle, bd. xxiv. p, 69) derives a formula for the expression of 

 the sum of /m integrals \//(a;i) + i^(.'C2)+ • • • -l-"4'(-*-'V)> ^^'here \p{x) has either 



the form -^ ^ ' y™ (a;) dx or -iLJ^ -^^ , where y{.x-) is an integral function 

 J X — cc J X — ct y ( '*^ ) 



of X, m a positive integer, and y"' (x) is one of the n roots yi (■>■), y^G'^")) • • 

 y„(.^■) of the equation 



o=2}o+2hy+p-2y'^+ • ■ • Pn-itf^+y^- 



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



The variables a;,(-i=l, 2, . . . ^) are the points of intersection of this 

 curve with a second algebraic curve. 



Rosenhain (Crelle, bd. xxviii. p. 249, 1844) employed as fundamental 

 equation 



<l>{x,y)=Poy''+2hr~^+ ■ ■ ■ +Pn=0. 

 in the place of Abel's ;:^(a;,y)=0. 



Proceeding in a manner very similar to that of Abel, he adopts in his 

 summation-formula integrands of the form 



Q-2r-^+Q3r-'+ ■ . . + Q" 



where the Q's are the rational functions of x. 



In discussing the hyperelliptic case (n = 2) he gives <t> {x,y) the form 

 /'oy2+7^iZ/+7'2> 3.nd not the form usually adopted, y"=^{x), p^, 2)1, 2^-2t 

 and R denoting rational functions of x. 



He then seeks to prove in the general case that the number of arbit- 

 rary constants in an integral of the first kind is equal to the smallest 

 number of integrals through which the sum of any number of such integrals 

 is expressible. The article is continued (Crelle, bd. xxix. p. 1). 



(20) Boole, in a paper, ' On the Comparison of Transcendents, with cer- 

 tain Applications to the Theory of Definite Integrals ' (' Phil. Trans.' 1857, 

 p. 745), contemplates the following objects : 



First, the demonstration of a fundamental theorem for the summation 

 of integrals, whose limits are determined by the roots of an algebraic 

 equation. Secondly, the application of that theorem to the comparison of 

 algebraic transcendents. Thirdly, the application of the same theorem in 

 a new, and, as it is conceived, more remarkable line of investigation to the 

 comparison of functional transcendents. In the introduction to this paper, 

 Boole states : ' As presented in the writings of Abel and of those who 

 immediately followed in his steps, the doctrine of the comparison of 

 transcendents is repulsive from the complexity of the formulae in which 

 the general conclusions are embodied.' With the intention of simplifying 

 these formulae, Boole introduced a symbol difiering in interpretation only 

 by the addition of one element from the symbol used by Cauchy in the 

 ' Calculus of Residues.' 



This symbol, (m^, he defines as follows : ' If (j){x)/(x) be any function of x 

 composed of two factors (p{x) and /(x), whereof <p(x) is rational, let 

 6[<l>{x)}/ (x) denote the result obtained by successively developing the 



