516 Mr. R. C. Rowe. Memoir on AheTs Theorem. [June 10, 



The generality and the power of this memoir are well known, but 

 its form is not attractive. Boole, indeed, in a paper on a kindred 

 subject (Phil. Trans, for 1857, pp. 745 — 803) says, "As presented 

 in the writings of Abel . . . the doctrine of the comparison of 

 transcendants is repulsive, from the complexity of the formulas in 

 which its general conclusions are embodied." Boole's theorems, how- 

 ever, escape this charge only with loss of the generality which makes 

 Abel's valuable. 



But this complexity is rather apparent than fundamental. It is 

 attempted, by rearrangement of parts, by separation of essential from 

 non-essential steps, by changes of notation, in particular by the 

 introduction of a symbol and a theorem discussed by Boole in the 

 paper already referred to, and by the addition of examples of the pro- 

 cesses and results, to reduce this part of an important subject to a 

 shape more simple, while no less general, than the original. 



Of the three sections which compose the paper the first contains the 

 discussion of the main question, 



Is it ahvays possible to establish between the values for different vari- 

 ables of the integral of an algebraic function however complex, algebraic 

 relations, the variables themselves being connected by any requisite alge- 

 braic laws ? 



If, for example, fXdx=¥(x), 



where X is an algebraic function of x, rational or irrational, in- 

 tegral or fractional, is it necessarily possible by connecting x±, x%, 

 . . . x n by any requisite algebraic laws to obtain an algebraic (or 

 logarithmic) expression for the sum I , (a3 1 ) +F(# 2 ) + . . . + F(a? n ) ? 



This question is suggested on the one hand by such well-known 

 results as 



lYah) +-F(aj 2 ) = constant, where X=— j== if aj 1 2 + a? 2 2 =l, 



Vl— x z 



and F(^)-fFfe) + r(^) = where X=-y====== 



v 1 — x* . I — k*x 4 



if (1 -xfl (1 -av>) (1 -x*) = (2-x^-xJ -xi + &V* 2 V) 2 ; 



and on the other hand by the possibility of finding algebraical ex- 

 pressions for many symmetric functions of the roots of equations, 

 though these roots may not be separately determinable. 



This combination of the theory of integrals and the theory of equa- 

 tions furnishes in fact the key to the problem, as enabling us to 

 express the " requisite algebraic laws " very concisely by a single 

 equation of which the variables are roots, and whose coefficients are 

 not independent but connected by a corresponding number of re- 

 lations. 



