34© Mr. Bromhead on the fluents of 
Cor. 2, This form may be extended to all the former 
Propositions. 
Cor. 3. As it is very tedious and often impracticable to find 
x in terms of v, in order to know whether the reduction be 
applicable ; the following process may sometimes be useful. 
Let the expression be 
dx . DR (x) . R”"” 1 R (x). R R(x) 
Z Z Z 2 
Then if it be divided by dx . DR (x) . R *R (x) the quo- 
% 12 . 
tient will be a rational function of R (r) or of the form 
a. + « . R(at) -J- a, . ^R (#)^* -f- • • 
— - * — the coefficients being indeterminate. 
a + a . R(x) + a . ( R (x)\ z + • . a 
o 1 a a \% / 
If the reduction be applicable, these may be found, and the 
substitution made at once. 
Cor. 4. We may thus reduce 
I . , n — 2 
nx + (« — x)c.jr 
z 
m V n I „ « — I , 
v x -f cx + . . . 
. R ^ x n ”|” c x n 1 -{- . . . j- 
R (z>), which may be found. 
Cor . 5. In dx . x rn 1 . [otx n -J- jS)^ . R | ax n + /3 j 
divide by dx . noc,x n ~~ l . (a,x n -j“ /3)^, and the quotient is 
~ . R [ux n -f /3). The latter factor is already of the 
required form, and by assuming 
x(r- 0 * = a . (*x n + Z 3 )" -1 + a • + /3 y- 2 4" • • 
the indeterminates may be found. In particular cases there 
are readier processes, but this method is universally applicable. 
