QUADRATURE of ‘the CONIC SECTIONS, &c. 297 
By: epee transformed an in the feries, it 
becomes: , ie 
i opunifionai yer. gard wxoK 2 hegre 
wast coe ee Cola A a Ded 
_ $e adeacearh cso esvedbe’ 4: 13—sof A — 1a ¢0f SA 
16; -3-+col2zA+4colA, far 3+colA+ 4coiZA 
4 ty 13 —‘cof* A —tr2cof+ ea) &e. t 
16° iia ike A+ 4coitA 
eee fer = <q be now fabftituted for A, and $a for + A, 
2 
and fo on, and let the refult be divided by 3X16; then we 
have 
gy Otedy 13— ofa + tacofpa — 
b yehe 16° 3+cola— 4colta 8.8.9.10 
% -13—coft a—12cof+a 13—cof+ Beacon 
PE a re cae Pee eR ery wea cots a 
I 13—cofya—12col ja 
ro 3.16° 3+ colga + ny oe aL Be. fs 
ia toe 16: 
which is our fourth, general feries for the rectification of an 
arch ; and its: rate of convergence.is very confiderable, for each 
term is lefs than, ;',th of the term before it. The feries, how- 
ever, approaches continually to a geometrical progreffion, of 
which the common ratio is ;';. 
29. Fue preceding formule, as well as innumerable others, 
which, may in like; manner be deduced: from the expreffion 
tf ee besos 
igh i 
tan A T tan A, all a ree 1n €x reflin a power 
atan yA ia gree i =P § & power 
of the reciprocal of an arch by an infinite fei the terms of 
which. 
