MR. R. C. ROWE ON ABEL’S THEOREM. 
743 
expression may be found for the sum of any number of such functions each multiplied 
by any rationed number positive or negative, integral or fractional. 
If all the rational numbers are positive and integral the theorem follows at once by 
supposing the functions whose sum we have shown how to express to be equal in sets. 
And this suggests the method of treating the general case when the numbers are any 
whatever. 
Let 9=g — a=fixed number to which the sum of the functions has been shown to 
be reducible. 
Then, by previous work (compare pp. 731, 732). 
*/']0L)+Vb( X 2)+ • • • f X l J a{ X a)=V— . . . J r(^ a + g)(x a + g )} 
t / , i(X 1 ) + ^o(X 3 )+ . . . +^ a '(X a -) = Y— {i/v +1 (X a , +1 )+ . . . J r\jj a - +e (X. a . +0 )} 
where a and a are any numbers whatever ; x a+l . . . x a+g are functions of x 1 . . . x a ; 
and X a , +1 . . . X o/+0 of X, . . . X a , and v, V are algebraical and logarithmic functions. 
Subtract: and let the last 6 of the terms on the left-hand side of the second be 
(both as to functional form and variable) identical with those in the bracket in the 
first. Then, writing (3 for a'—6, we have 
M x i)+ ■ ■ • -b^aK)—^(Xi)— . . . — ipp(Xp)=v—' V-b{t// a / +1 X a+1 + . . . -\-xjj a+e (X a+e )}. 
Equate all the functions on the right to zero. 
This will give 9 relations between the cc’s and X’s. 
Now making the functions on the left equal in sets, and dividing by any requisite 
integer, we have a result which may be written 
~b ’ • ' m) = Y7 
where the fs are similar functions, m is any number whatever, W is an algebraical 
and logarithmic function of the y s, which are themselves connected by 9 relations, and 
the h’s are any numbers wdiatever. 
If we express 0 of these variables as functions of the rest and call them zs, putting 
n for m — 0, we can write 
J r • • • fh r f n (y n )=iv-\-Ic l <f)' l (z L ) + . . . -\-k e <f) e (z g ). 
Or making, as we may, the Jc’s each=unity we have shown how to find the 
expression required. * 
* The subscript letters attached here, and not before, to the functional symbols introduce no novelty. 
They are only intended to suggest the fact that what we have written t/'■( a a), • • • are rea Ty VYi’ V\)-> 
p(x%, y%), . . . ; while y 1 and y 2 . . . are not necessarily the same functions of Xj, cr 2 . . . This has not 
been hitherto overlooked, it is only more clearly put in evidence now. 
