Spherical Motion. 
5 43 
a 
s/jr + ® - q \Zir~ 
X 
2 
but the time or motion can never 
\t~® 
begin at B, nor can the pole oppofite to B ever come under 
Z. And the reafon of this is alfo evident from the nature 
of the motion itfelf; for thefe being poles of a permanent 
axis, if Z were once over one of them, it muft always con~ 
tinue fo. 
Having thus determined the time, velocity, and manner, 
in which the fpherical furface that revolves with the body 
pafies under the fixed point Z, it only remains to determine 
the path of one of the poles as C of the permanent axes about 
Z in abfolute fpace, or upon a fpherical furface at reft, but 
equal and concentric with that fuppofed to move with the 
body ; for the path of one of thefe poles as C being found, 
thole of the other two, and indeed the path of every other in- 
variable point of the moving fpherical furface, becomes known. 
Now, the velocity with which C approaches Z is found above 
f cz 
•b 1 Xxy 
and the fluxion of the arc CZ 
col', cz 
v'aW + b y ’ 
T ~ cz ~ by the velocity gives /, whofe fluent is found 
above, and confequently the diftance of C from Z at the end 
of any time /, there is then only wanting the angle defcribed 
by C about Z, correfponding to the diftance CZ therefrom, to 
have the path of C about Z ; which may be found by the 
help of quadratures as follows,, 
As f. ZC : velocity of C perpendicular to ZC (found above) 
* x cof. AZ +y x cof. BZ , , . . _ _ 
— — 2c :: 1 : the angular velocity of C about 
% x cof. Az. "•}- v x cof. 13 Z i . , , . . • .... 
/j = Czc 1 ” which velocity being multiplied by 
cof. 
