113 
If y is expressed as an explicit function of x, the expression will contain, in 
general, a root of some polynomial. The definition is sometimes stated as follows: 
fr (a, y) dx, 
; m 
y=V Rn (2). 
Byerly (Integral Calculus Ch. VI) observes that ‘‘ very few forms (of the above 
type) are integrable, and most of these have to be rationalized by ingenious sub- 
stitutions.” It is the purpose of this paper to determine the conditions under 
which irrational algebraic integrals can be reduced to rational algebraic integrals 
and to present a method of obtaining a substitution by which the integral is 
rationalized. 
Given then, the integral : 
fF (a, y) dx. 
fu (2, y) =0, 
where F is a rational function, to determine the conditions under which this in- 
tegral can be reduced to the integral: 
A (2) da. 
Let us consider in the first place the integral : 
fF (a, y) dx. 
yo=V (BMG): 
where R (x) is a polynomial of the second degree. 
The equation : 
(ats) gh (ay==o 
is of the second degree and hence represents 
a conic section. Let A be any point on 
the curve and let 
where r is a rational function. 
(2.) Bod 05 
be the equations of any two lines through 
A, Then 
(3.) ptaq=o, 
represents any line L through A. Since 
this equation is linear in x and y it can be solved for y by rational processes. Let 
the solution be: 
(4.) p=? (a 2). 
(5) = 9? (2, 2). 
(6.) -*. 02 (a, 4) — R (a) = o. 
8—SCcIENCE. 
