irrational functions. 
j dx . R | x, s/ a xd -f- / lx -{- 7, (ax -f* s/ a? x* - {- [lx -J- y) m , ..... | 
with an indefinite number of forms too complex for conve- 
nient expression. 
Cor. 3. This form may be extended to Prop. II. and other 
general expressions. Thus we know 
1 
n 1 
The forms given above are wholly inapplicable, when the 
fluxion involves expressions, such as R^R R“ s ... (x) where 
1 t 1 
the functions are alternately inverse and direct. The cases 
are very few, in which the difficulty can be overcome, and 
perhaps the following Propositions will be found to include all 
the instances, in which analysts have effected the reduction. 
Prop. IV. 
We can rationalize 
dx . DR (x) . {R(x),R~ j R(x) } 
a '-a 1 a ' 
Let R (x) = v, and it becomes 
£ 
dv . R[p, R \v) } as in Prop. 1. 
Cor. 1. We can generally reduce Jdx . D<p (x) .pep (x) to 
Jdv . <p (v). Thus we deduce from Jdx . x rn ^\ a? (x n ) the 
fdv . v r ~~ l (p (v), and fromJ~.(p (x n )thej^] q> (v), reduc- 
tions of frequent occurrence, by which analysts have given 
their forms an appearance of generalization without the reality. 
mdcccxvi. Y y 
fdx.R ( x, R - '(x), R -1 R _, (x), .... (R~’..R -, (x)) 
X 1 1 a 1 n 1 
