Spherical Motion . 
iv. the momentary pole fhxfts its place along its own track 
; and if Z be fuch as that the great 
with a velocity = 
circle OZ (fig* 4 - ) rosy always be perpendicular to the track 
WV'W'V (fig. 8.) that paffes under Z, which it muff be if 
O be the pole of motion ; then as i : :: f. OZ 2 =cof. 
the fquare of the velocity of the track under Z = 
nr!A2i Zt tL 
~~~ , hence f. OZ 2 =z — * — — - 
AC 7 BV 
ACa 
1 9 
cof. OZ : 
B 
i 
ACfa 2 
j«ft 4 a a C* ■>/ 
“ I *j“ — 
a- AC 
B 2 2 
> 2. .2 
B 2 a 
£ 
O 
m z b 4 zl z $L z tfAl 2 ' -f t: z C 2 | 
BV 
C0 f.ZO= ^ -+ ^ = f. OY; this, then, is the value of 
cof. ZO deduced from the fuppofition that it is always per- 
pendicular to the track upon the moving fpherical furface 
which paffes under Z at reft ; and if this be found to agree 
with the value thereof computed by trigonometry, it will 
prove the legitimacy of that fuppofition, and that it is the true 
value fuch as that O fliall be always the pole of the momen- 
tary axis and Z at reft in abfolute fpace. Produce BZ (fig. 4.) 
till it cuts AC perpendicularly at q ; then it is before found, 
that the fine and cofine of BZ are m\/a 4 x z -f- c V aud w^,thofe 
of CZ ms/ a V 4- b A y z and mrz, and as f. BZ : 1 :: cof. CZ : 
& z X a 4 d 2 -f- c 4 €^ 
and 
f. AQ 
c z % 
and cof. Aq = 
a x 
^ a 4 x z -f- 
y ^ 
= cof. BO, 
= f BO, * = cof. CO, = c CO as f BO : i :: 
cof CO : f QA 
, cof QA — -- A. , and the arch Qa 
V x 1 -f Z z 2 
= AQ-A^ — the meafure of the angle OBZ, and cof Q^ = 
X Z + z‘ 
Vol. LXXX. 
4 C 
