Spherical Motion. 517 
are f. A'Z and f. EZ lifting their places in abfolute fpace, 
which therefore can in no wife affeft the velocities round thofe 
centres, which velocities muffc ftill be the fame relatively to 
the centres as if the centres were at reft. Hence, then, the 
nature of this fpherical motion is fuch, .that the axis whofe 
pole is Z being abfolutely at reft, the pole O fo fhifts its place 
in a circle whole radius ZO alfo at reft, as to do fo with a 
conftant velocity = eh x the velocity with which it 
J i. AZ~~ A 
fhifts its place in the circle eO on the moving furface, the 
track therefore on the moving furface oiculates or rods upon 
that on the immoveable one. Therefore, fince — = the time of 
one revolution of O upon the moving furface, and the time 
of one revolution of A' 3 and confequently O round Z = , 
.. -- ; in the time of one revolution of O on the 
moving furface, it will have fhifted its place round Z in the 
circle whofe radius zzz f . OZ, through an arc — the whole peri- 
phery x + ? that is, it will have made + aA + 1 
o'* 
revolutions round Z : for, as the two circles eO and EO ofcu- 
late, it will take + 2 A + 1 times the periphery of 
EO to go round eQ, that is, the point A\ and confequently O 
will have moved this number of times round Z at reft, whilft 
O fhifts its place once round the fpherical furface in motion. 
Hence then the nature of the motion round the momentary 
axis whole pole is O, and the fixed one whofe pole is Z, will 
be apparent from the following iimple contrivance. A circle 
EO to radius = f. ZO-f. ZE being drawn upon a v fpherical 
furface at reft, an odtant of which is ABC, let a paper, or 
other 
