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XXV. On the fluents of irrational functions. By Edward 
Ffrench Bromhead, Esq. M. A. Communicated by J. F. 
W. Herschel, Esq. F. R. S, 
? V: 
Read June 4, 1816. 
V JfSWQO; 
Th e efforts of analysts in determining the fluents of rational 
functions, have been completely successful, and their labours 
form one of the most perfect and beautiful branches of the 
fluxionary calculus. In the irrational functions, however, we 
find but little effected. With the exception of Waring, 
modern analysts have not added any thing important, to the 
forms given by Newton, Craig, Cotes, and Bernoulli. 
No attempt has been made to generalize the known forms, 
and the last eminent writer on the subject. La Croix, seems 
to consider them as independent results, not deducible from 
any common principles, and refers us to the Petersburgh 
Acts, and other miscellaneous Collections. In the following 
pages, it is attempted to generalize and systematize our know- 
ledge on this subject ; and to show that all the known forms 
result from other forms of the greatest extent, not depending 
on particular functions, but upon the properties of all rational 
functions whatever. 
R, R, R, R, denote rational functions of any kind ; R"” 1 , R~" E , 
s a n 1 
R \ R"* 1 their inverse functions. Thus if ^ = R (^) any 
*2. n 
rational function of (z>), then v = R —I (x) the inverse 
function. 
