Spherical Motion. 501 
motion on the furface may vary ; but whatever point on the 
furface correfponds with that pole muft at the inftant be at 
reft. 
PROPOSITION III. 
Let ABC (fig. 3.) be an o&ant of a fpherical furface in mo* 
tion, while the centre is at reft ; and let the velocity of the 
great circle BC in its own plane = a, and in a fenfe from B 
towards C ; that of C A in the fenfe from C towards A — b, 
and of AB from A towards B = c. If thefe three velocities 
a, b t and c, be conftant, the fpherical furface will always re- 
volve uniformly about the fame axis of the fphere at reft in 
abfolute fpace. 
For, let ABC, abc, be two pofitions of the revolving o&ant 
indefinitely near each other, A a, B b, and C c, the tracks of A, 
B, and C, in abfolute fpace. Perpendicular to ha draw the 
great circle SO A, and perpendicular to B b the great circle BOQ, 
cutting SOA in O and CA in Q.; then, becaufe A a is indefi- 
nitely fmall, the two triangles A pa right-angled at /, and a' Aq 
right-angled at A may be confidered as plane ones, and are 
therefore fimilar; and fince the angles pAQ and qAa are both 
right ones, taking away qAp, which is common, the angles 
pAa , qAQ , muft be equal; but as pA : pa :: c : b y likewife 
pA : pa f. pa A : L pAa, and paA=pAq 9 pAa — qAQ\ con- 
fequently, as f. pAq : ft qAQ c : b 9 that is, the fines of the 
angles BAS and CAS are proportional to the velocities along 
AB and CA; confequentlv, the fines of the arches SB and SC 
which are the meafures of thofe angles muft be in the lame 
ratio. In like manner it appears, that as ft CQ : ft AQ :: 
