Spherical Motion . i 533 
the fixed point Z mud be== — c:>f ‘ r J'^ c S£. z x ; in like man- 
* X cof. ZC — % x cof. Z A , , - ^ 
— — the velocity of B along BZ, 
ner is found 
and 
f. ZB 
_y x cof. ZA— * x cof. ZB 
~ 57zc 
= that of C along CZ in abfolute 
fpace. But the fluxions of the arcs ZA, ZB, and ZC, are 
• • * 
cof. ZA cof. ZB cof. ZC r ri . , , . , 
T~zaT 5 ”Tzb~’ and ' fz ' c * re lp e ccvely, which divided by 
their correfpondent velocities, give the fluxion of the time, 
cof. ZA 
that is, i 
z • 
a : 
z x coi. ZB — x cof. ZC z^j —yc*z 
cof. ZB 
(above found) 
cof. ZC 
x X cof* ZC -2X cof. ZA xc z z—za z x y x cof. ZA~~x x cof. ZB 
— - — from which fix- fold equation, it is evident, by in* 
ya x — xby x J 
fpedlion only, that if m~- any conftant quantity whatever, and 
ma z x = cof. ZA, mb z y = cof. ZB, and == cof. ZC, all the 
conditions thereof will be anfwered. Then, fince cof. ZA 2 -f 
cof. ZB 2 4 - cof. ZC 2 = i , its equal m z a v x z 4 - m z b 4 y z m z c 4 z z mull 
alfo be — i : but from the former part of the procefs 
a 4 x z + b 4 y z + c 4 z z = cftyp -f- b 4 $5 z + c 4 € z ; therefore m = 
sr a conftant quantity ; and f. AZ 2 = i - cof.AZ 2 
cof. BZ 2 -f cof. CZ 2 = i — m z a 4 x z = m z b 4 y z -f m z c 4 z z 9 f.BZ 2 = 
i — cof. BZ 2 — i — m z b 4 y z — mra 4 x z -f tn z c 4 z z , and 1 . CZ 2 = i — 
w 2 cV = m z a 4 x z + ^ 2 $ 4 jy 2 ; and, from above, the velocities with 
which A, B, and C, approach Z are refpe&ively 
; but as a is fuppofed 
b z — c z xyz f — x xz 
and 
a — b % x xy 
VTjfZjT ’ v * 4- c 4 z z 7 V aXx z 4- by 
greater than c 9 c z - a z is negative, and the velocity therefore 
in a contrary fenfe, confequently the poles A and C mull 
approach Z, whillt B recedes from it. The relpedive veloci- 
ties 
