5°P 
Mr . Wildbore on 
PROPOSITION II. 
Supposing the centre of a fphere to be at reft, whilft the 
furface moves round it in any manner whatfoever ; then, if the 
fame invariable point O, confidered as the pole of an axis of 
the fphere, be itfelf in motion, the angular velocity of the 
fpherical furface about that axis will be unequable, or that of 
one point therein different from that of another. 
For, let I (fig. 2.) be the centre of the fphere; draw the 
great circle POF perpendicular to the direction of the motion 
of the furface at O; then mud the pole of this motion necef- 
farily be in fome point P of this great circle POF. Let FC 
be the great circle whofe pole is P, and LQ that whofe pole is 
O ; then, the velocity of any point F of the great circle FC 
muft, by the preceding propofition, be equal to that of any 
other point H thereof. Let that velocity be reprefented by the 
equal arches FG and HK, and from the pole O draw the great 
circles OGM, OHN, and OKA, cutting the great circle LQ^ 
in M, N, and A ; then muft LM reprefent the angular velo- 
city of the point F about the axis IO, and NA that of the point 
H. But, by Prop. 9. Lib. III. Theodosii Sphericorum, LM 
is greater than NA ; and confequently the angular velocity of 
the point F about IO is greater than that of H ; and confe- 
quently the angular velocity of the furface about the axis lOis 
unequable. 
Corollary . Hence, about whatever axis the angular motion of 
a fphere is equable, the pole of that axis, and confequently 
the axis itfelf, muft be at reft at the inftant. Different motions 
may have different correfpondent poles, and confequently, 
when the motion is variable^ the place of the pole of equable 
motion 
