Mr. Wildbore on 
is O; let this angular velocity — e, then mull ext S 0=a, 
ext. OR = s, and e x f. GQ=y, and the hum of the fquares 
of thefe three, or a 2 — e 2 x 1. SO + e x f. RO ! + f x l 
f.QQ z = e 2 , becaufe f. SO' + f. RO + f. QO =i, hence e = 
^/a" + z* +y z ; whereas, before the impulle e- ^/a r 4- b z + r. 
Thus not only the place of O, but, if z +jy be not = S + c , 
the angular velocity of the fphere about its lingle axis will alfo 
be altered. Hence then if, inftead of an inftantaneous im- 
pulfe, a motive force be fuppofed to ad in the fame diredion, 
and meafured at the fame point where the impulfe was juft 
now fuppofed to ad ; fuch force can neither vary the point S 
nor the velocity a, but will in time vary b and c, and caufe the 
point O to alter its place in SA ; and thus the velocities b and c 
will vary to y and z, and e — + b + c tor — \/ ’ & -ty + s » 
juft as it would have been by a fingle impulfe, excepting that 
then, when the impulfe was over, y and z muft have become 
conftant quantities, whereas now they will vary perpetually 
during the time that the motive force ads, and the point O 
will fhift its place fo as at different times to coincide with dif- 
ferent points of AS, though at any one inftant the point 
of the furface that coincides with it muft be at reft, by 
Prop. 2 . 
PROPOSITION IV. 
If a fpherical furface, whofe center is at reft, revolve in any 
manner whatfoever, fo that the velocities along the three qua- 
drants bounding any odant thereof be exprefled by any three 
variable quantities x, y, and z ; to find the neceflarily cor- 
refponding accelerating forces with which the place of the 
natural 
