732 
MR. R. C. ROWE ON ABEL’S THEOREM. 
x i • • • x a being any quantities whatever, in terms of an algebraic function and a 
number of functions of the same form whose variables are themselves definite functions 
of the quantities 
The question then arises, What is the smallest number of functions in terms of 
which the sum may be expressed ? and can the sum of any number of functions be 
expressed in terms of this smallest number 
13. Required the least value of which the difference between the number of roots 
possessed by the ‘equation of the limits’ and the number of constants introduced by 
the ‘ equation of condition ’ is susceptible. 
This difference is expressed by /x— a. We must put each term under a different 
form. 
(i.) For a. 
Let us express the index of the highest power of x in a function 3{x), supposed 
rational and integral, by the symbol J (x). 
Then m general the number of coefficients in J (x) is J( x ) -fi 1 ; and as in 6 one 
coefficient may without loss of generality be written unity 
«— number of coefficients in 0(=q n _-{y n ~ l -\- . . . +'/,,) 
= Sq-fi-n— 1. 
Two corrections must be introduced. 
For the existence of each linear factor of F 0 implies a linear relation between the 
as, and diminishes the independent number by unity. We have on this account to 
subtract F 0 . It may happen, however, that the particular form of the function 
renders the number of necessary relations less. Write then F 0 —A as the quantity to 
be subtracted. 
Suppose again that some of the constants are so chosen as to reduce the degree 
of E.t 
In general /x and a are thus equally reduced ; but it may happen that the form of 
the function renders necessary a less number of conditions. If this lessens /x by a 
number greater by B than the lessening of a we have to use instead of F 0 —A, 
F~-A-B. 
We will however for the present drop the A and B, which would appear without 
alteration throughout the process, and replace them at its conclusion in the shape of 
a correction to the result. 
* In an earlier memoir (Abel’s works, vol. ii., xi.), this question is dismissed with the remark “ il n’est 
pas difficile de se convaincre que, quelque soit le nombre /t on pent toujours faire en sorte qne n —/i 
devienne independant de Here the actual value of this constant is investigated. 
t For example, in the case of p. 725, we put \/l — .r 2 .l —c 3 a: 2 = 1 +px + r/.r 2 , and the assumption of unity 
as the first term on the right reduced the resulting equation from a quartic to a cubic. 
