irrational function s . 
34 1 
Prop. V. 
We can rationalize 
dx . R fx, R““ ! R (x) 1 if we can determine, IC^R (x) = 
1 - 1 n ' * n 
R^.R-^x), for the fluxion then becomes 
dx . R$x, R(x) . R“* J (u:)|as in Prop. I. Cor. l, 
s> a 
Cor. l. We also know 
dx . Rf x, R-‘ (x), R— R (x) 1 
la t n J 
Cor. 2. We may thus transform 
«*+0j ”,V(^x + / 3 ) . ( ax + b) n ~‘} 
‘ ^ { X > (« + b 
into 
dx . R|x, 
a known form. 
Hence we know 
. («x±£)} 
fdx . R jl •£» l/(x 4” ^ ) • (-T + /3) 2 1 
. R| x, V («.r -f- / 3 ) . + 6)| 
Jdx . R | x, CX 1 c x 4” c ^ 
which last form will sometimes introduce imaginaries, that 
may be avoided by particular artifices. 
Cor . 3. If R— 1 R (x) = (R (x)\ m . R"" 1 (x) 
R~ 1 R(x)= |R (xj]».R-' I (x) 
&c. = &c. 
