2 54 DEVELOPMENT of a certain 



as the feries which moft readily occurred to them, converged in 

 fome cafes fo flowly as to be in a manner ufelefs, no fmall de- 

 gree of analytical add refs has been found neceflary, either to 

 render it more convergent, or to find the fum of a competent 

 number of its terms, with a moderate degree of labour. 



2. But in confidering the fubject, it has occurred to me, that 

 although we cannot exprefs the values of the coefficients in 

 finite algebraic terms, nor even by means of circular arches, 

 or by logarithms, yet when n is the half of an odd number, ei- 

 ther pofitive or negative, we may always exprefs them by means 

 of the proportion which the perimeters, or femi-perimeters, of 

 two ellipfes bear to thofe of their circumfcribing circles. The 

 problem may therefore be reduced to the rectification of the 

 circle and ellipfe, and mathematicians know that fuch reduc- 

 tion is confidered as the next degree of refolution, in point of 

 fimplicity, to our being able to effect the folution by means of 

 circular arches, or by logarithms only. 



3. It is well known that we can eafily obtain a fluxionary ex- 

 preffion for each of the coefficients A, B, C, &c. in the equation 



(#2 _|_ £ — 2 ab cofcp)" = A -f B cof<p -J- C cof 2<p -f D cof 39 -f- &c. j 



for if each fide of the equation be multiplied by <p, and the 

 fluent taken, we get 



/^^ + ^_2^cof<p) ; ' = A(p + Bfin<p-r-iCfin2(? + jDfin3(p+&c, 



Let us now take the fluent generated, while <p from o becomes 

 a femicircle, then fin <p, fin 2<p, fin 3<p, &c. all vanifh ; fo that, 

 putting * to denote half the perimeter of a circle, of which the 

 radius is 1, we get, 



a-A =/<p(a 2 + b 2 — tab cof <p)". 



In like manner, if we multiply each fide of the affumed 



equation 



