34)2 Mr. Bromhead on the fluents of 
we can determine 
-l JL 
fdx .k] x,R~' (x)A R~‘ R (*) V\- \ R — 1 R (x) } ” , . . } 
Cor. 4,. If R~‘ R(j) = R (x) V R- 1 (x) ; R — 1 R ( x) — R (x) . 
* m (A &, •x n , 
n ^W~\x ) ; &c. = &c., 
; a, 
we know 
fdx . r{ x, R-' (x), R _I R (x), R-‘ R (x), R~‘R (x), . .. } 
'■/v" t m * n 3 r 3 
as in 
dx . R | x, •sA ax -j- ( 3 ) . (ax b) m 1 , V( xX + f^Kax-\-b) n - I J 
Cor. 5. Generally if R (a;) and R (i) are so related, 
1 a 
that R R (a:) can = RR (x), R and R being any rational 
in x m n m 
functions whatever taken at pleasure, then dx . R | x, R —I R(x) J 
can be rationalized by taking x = R {v). It then becomes 
m 
dv . DR (v) .R {R (w), R (?) }. 
Prop. VI. 
By combining Prop. IV. and V, we can rationalize 
dx . DR (x) . R {R (x), R” 1 R R ( x ) J 
. . -, Vl m m 1 » m * 
if R— 1 R ( v ) = R (v) . R— 1 (v) ; for let R (x) = v, and it 
1 n y z m 
becomes 
dv . R | w, R (^) . R” 1 ( v ) | as before. 
v 2 
Cor. 1. If we have p 
dx . xn—i (oiX n + ( 3 ) ? . R | a -f & % ~~ n } 
