Spherical Motion. - >54$ 
ZS = I , with which radius on the centre Z defcribe the circle 
STR' on which take ST = any value of A, and through T 
draw ZY = the ordinate correfponding to that value of A, and 
thus may points at pleafure be found, and tne curve GY 
conftruded. Now, when ZC = CV, the value of ZY = 
ba 1 x f. ZC*-$j*x 
6 — £ 
_ — — ■ is infinite, and if SR~' 
V^X^-TXf. ZC 2 V'LC\' z -uZC z 
the then value of the arc A, ZR produced will be an afymp- 
tote to the curve GY. But to remedy this inconveniency 
anfing to the conftrudiion from this infinite lengtti 01 the cuive , 
produce any other radius ZR of tne circle, till ZO — the firff 
value of ZY 7 \J - 
* a 
ba*yj. ZC* - h' 1 ya-c 7 
V'V — Z x a —O' xf. ZC X cof. ZC vV — cof. CZ* 
when CV = ZC and the arc B = o, and taking ZY = any 
other vali/e thereof correfponding to fome value R'T' of the 
arc B lefs than RS' the value thereof when n - cof. ZC and 
ZY' infinite ; and thus the curve HY may be conftruded by 
points ; let the conftrudions of both thefe curves GY and HY' 
be continued till the value of the arc ZC in the one conftruc- 
tion be equal to that in the other; then muft the lum of the 
correfponding areas ZGY + ZHY' be equal to the infinitely 
extended area formed by each curve running out towards its 
own afymptote, each of thefe infinitely extended areas being 
equal becaufe they begin together, and are the fluents of the 
equal fluxions Ax^andBx ~ Equal to any value of the 
area ZGY, let the fedor QZR be cut off from the circle whofe 
radius is unity ; then the area of this fedtor ~ half the arc RQ = 
cof. ZC 
the fluent of Vra rzc> 
Vol. LXXX. 
X 
2 a 
Y >_ £ : 
4 B 
ba' xf. ZQ?-b?- a z -c’- 
f.ZCVi.CG-l.ZC* ’ 
and 
