JS54 DEVELOP MEN't of a certain 



as the leries which moft readily occurred to thein, couverged in 

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

 gree of analytical add refs has been found neceffary, 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. Bur in confidering the fubjecl, it has occurred to rne, 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 redification 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 fokxtion by means of 

 circi^lar arches, or by logarithms only. 



■J. It is well known that we caneafily obtain a fluxionary ex- 

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

 ^(i^ _{- b' — 2ab cof?)" =: A + B cof(? + C cof 2(p -f D cof 3?) + &c. ; 



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

 fluent taken, we get 



/^(^^._|_i. — 2rt^cof(p)" = A<p + Bfin(p-f^Cfin2$-|-jDfin3f+&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 vaniffi ; fo that, 

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

 radius is i, we get, 



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



equation 



