QUADRATURE of the CONIC SECTIONS, Sc. . 293 
, 26.’'Txe mode. of reafoning by which we have. found, {eries 
rioiepme ings the three finft powers of the; reciprocal of an..arch, 
 will-apply, equally to/any higher power, but)the feries will be- 
‘come more and more complex as we proceed, befides requiring in 
their application the extraction of high roots. In the cafe;of 
the fourth power, however, the feries is fufficiently fimple, and 
converges fatter than-any we have yet “inveftigated, while, at 
the fame time, in its application we have only extractions of 
‘the fquare root. On thefe accounts, I fhall here give its invef- 
paestian. 
i I 
MoS oy I 
fit te ae ex effion So i 5 tan = A: 
ee tanA” 2taniA 2, 
let the fourth power, and alfo the fquare of each fide of the 
equation be taken, the refult will be 
eR set eh al = = 44 
tan¢+A 16 tan* +A tera nh3 eke 2A + S tan"+ A, 
IT “ I * I I Le 
a ian A — qamgA 314 tan*1+ A. 
a firtt of thefe equations be multiplied a me and the 
‘fecond by, 3, and let the refults be added ; then, reducing the 
fractions to a common denominator, we get » 
tan’ A YI 3+4tan’ tA 
en Crain ordeal ro + go tate sts tan’ A}. 
LET us, for the fake of brevity, exprefs the couples quanti- 
tan* A 
ftood as raed ‘produét of two quantities f and A, but as a ¢harac- 
ter denoting a particular function of the arch A ;) and, fimilar- 
ty tater Ay by the mbok 7 A, (which i is not to be under- 
hy, let Gan 2= be denoted by f $A, and foon. Alfo 
. / 
Oo 2 let 
