33$ Mr. Bromhead.ow the fluents of 
Jdw . R [x, a + y b + s/. . * x /\ R~‘ (x), . . } 
and some of the most complex expressions in Waring's 
M ed. Anal, are very particular cases of this form. 
Prop. III. 
We can rationalize 
*.r{*,r-(4 (r -»]=; (r~V))\(r-\*)|; . . . } 
Let R"”” 1 (jr) rrr v m • n .r . . 
Then x = R (v m • n • r • 
“ s 
= v n • r • * 
X 
{R- 1 ^) | n = v m * '' * • • 
&c. = &c. 
which substituted make the expression rational. 
Cor. i. The more general form is this : 
If R can be so assumed that R” -1 R, R ~ 1 R, R - ' 1 R shall 
m n m 
be all rational; then by assuming R l (x) = R (v) we 
i m 
can render rational 
dx. Rj.r, R-'(x), R— R— (x), . . . R~> R-'(jc)i 
Cor. 2, We can find 
i i i 
Jdx . R {x, x™, x~,x~, .. .} 
I I 
Jdx . R [ X, (aX + (3) m \ ( uX 4- P) n , . . . j 
jf dx . R | x , 
a.X -j" 
m 
ax + bj 3 [ax + b 
aX + |3 \ n 
J"’"-} 
