514 iM>. WiLDBORE on 
taking A y ^=:AE, the point O will have fhifted its place in the 
time /, in abfolute fpace from E to O ; and upon the moving 
fpherical furface from e to O along a lefs circle whofe radius is 
equal to the fine of AE = f. A!e~{. A O ~ b. Now, at the 
commencement of the motion, that is, at AB, the firft velo- 
city of the point A along CA is e x f. AE = eb = the then value 
of y, becaufe the pole of the natural axis of motion E being 
then upon AB, the value of 2 = 0, and the pale E fhifting its 
place in the fenfe EO in abfolute fpace, and the invariable 
point A of the fpherical furface moving in the fenle AA% there 
mu ft be fome point Z between E and A at reft with refpeft to 
both thefe motions, or round which both of them may be 
fuppofed performed ; its property muft then be fuch, that as 
f. AZ : f. EZ :• velocity of A ~eb : to the velocity with which 
jP !EZf 
the pole E begins to fhift its place = eb x ; but is the 
velocity with which it fhifts its place upon the moving fpheri- 
cal furface at E about the radius = f. AE m f. AZ + EZ. And 
when O is at E the velocity with which the fpherical furface 
pafles under Z will be e x f. EZ. Again, when O is the 
place of the momentary pole, the velocity of the point A' =z 
s/y -f % = eV y 4- <T = eb as before, and the velocity with which 
O fhifts it place round A' as before ; it therefore fhifts its 
place round fome point Z in abfolute fpace fuch that eb x f~£z, 
ftill the velocity with which it fhifts it, which, becaufe A O = 
V 
AE, muft be the fame velocity and the fame point Z as before. 
Confequently, the point O fhifts its place along a lefler circle 
of the fphere whofe radius ^ f. EZr=f. OZ, and in the time of 
fuch fhifting from E to 0 ? or from A to A 7 , the point of the 
2. furface 
^ ' 
