Spherical Motion . 503 
from B, C, and A, mull be always at the fame distances from 
O, that is, OR, OS, and OQ, are conftant, and the point O 
at reft. And this muft alfo be the cafe if a , b , and c be 
variable, provided they have the fame conftant ratio amongft 
themfelves. ' 
Corollary . Hence the points Q, R, and S, are the nodes of 
the great circles GQA, ARB, and BSC. 
Scholium . The demonftration of this proportion being thus 
ftriftly given, fomc notion may be obtained of the manner in 
which the point O 'varies its place upon 1 the fpherical furface 
when the velocities along the circles AB, BC, and CA, are va- 
riable. Thus, let fuch fpherical furface, fo revolving, receive 
an inftantaneous impulfe, at the diftance of a quadrant or go° 
from S, in a direction perpendicular to the plane of the great 
circle CSB ; then, the centre of the fphere may be kept at reft 
by an equal and contrary impulfe at this centre ; and finee, by 
hypothefis, the impulfe is given 90° from S, and in a direction 
perpendicular to the plane of the great circle CSB, it can nei- 
ther alter the place of the node S upon the circle, nor the ve- 
locity in the diredlion of its periphery, but only thofe in AB 
and CA. Thus, if the velocity in BA which before was = r 
be now equal to z; then, as f. SB : % f. SC : the velocity 
along CA, let this==y, whilft ftill the velocity along CB con- 
tinues as before™^ ; and this will caufe the point O to fall upon 
another point of the great circle SA : fo that whereas before 
the fines of OS, OR, and OQ, were as a, c 9 and b 9 they (hall 
now be as a 9 and y* Confequently, as f. SO : rad.n 1 a : 
the velocity at 90® from O, f. OR : 1 :: % : the velocity at 90° 
from O, and f. OQ :y :: 1 : velocity at 90° from O, which 
three quantities muft therefore be equal to one another, and to 
the angular velocity of the fphere about the axis whofe pole 
is 
