742 
ME. K. C. ROWE ON ABEL’S THEOREM. 
and by the definition of r 
[0]-[2]=2r 1 
while from the conditions (E) 
2t 1 >3—fa—A a , i.e., >3 — 5, or 3—f —\ 
<3—|-afi-A a , i.e., <3 
so that 2 t 1 >0<3 
whence [0]=[2], [2]+l, [2]+2, [2] + 3, 
and so if the degree of q. 2 be denoted by 0 that of q x is 9 + 1; and that of q 0 may be 
either 9, 9+1, 9+ 2, or 0+3. 
We have, then, by art. 13 (i) 
a: =[0]+[ 1 ]+[2]+2 = 36>+3, 30+A, 3 0+5, or 30+6 
while 
P =( M+PWi) + n stH{ M+ P-P *} 
= 2(0+3)+ {0, d + 1, 9 + 2, 0+3} 
= 30+6, 39+7, 30+8, 30+9 
So that, as on the last page, 
fx — a —3. 
We have proved then that the sum of any number of integrals of the form indi¬ 
cated by the fact that they are rationalized by the introduction of y, where 
y*+l+f+2+J+Po=°> 
can be reduced to the sum of three; the equation of condition being qyy~+qiy+qo— 0, 
where q x — q 2 + 1, and q 0 lies between q. 2 and q 2 +3 inclusive. 
Section III. 
22. We have shown that the sum of any number whatever of similar functions such 
as are discussed in this paper can be reduced to an expression algebraical or logarithmic 
added to a fixed number of such functions whose variables are functions of the variables 
of the given functions, this fixed number depending only on the form of function 
considered. 
From this a more general theorem may be shown to follow, viz. : that a similar 
