5 1 B • - Mr . Wil'd bo he on 
other loofe furface, be fitted to this ochnt, and having on the 
centre A and radius AE deferibed a circular arc on the loofe 
furface, let the part thereof EOFCBE be cut away, and 
completing the circle EF of the remaining part, let the cir- 
cumference of this circle be moved uniformly along the cir- 
cumference of the lefs fixed circle EO with the celerity 
^ beginning at the point E in each, fo that the moving circle 
may roll along the fixed one, that is, fo that the arc Oe of 
the moving circle which has been in contaft with the fixed one 
may be always equal to the arc EO of the fixed one with 
which it has been in contaft ; then, fince OZ and ZA 7 arc 
conftant, and OZ perpendicular to both circles, the point A 7 
muft deferibe upon the fixed furface, the fame locus as in the 
cafe of the motion above fpecified. The locus alio of the mo- 
mentary pole O will be the fame, and the angular velocity of 
A 7 about the momentary axis the fame as that of the moving 
fu rface about it : for the celerity of O about the axis whole 
poleisZ = ^ being equal to the celerity about A 7 in motion, 
and the locus of A ' being a circle whofe radius = f. ZA, we 
have, as f. ZO : f. ZA 7 :: ^ : ^ = the velocity of A 7 , and as 
f OA ' : eb :: radius = i : £=zthe velocity about the mo- 
mentary axis, as it ought. 
From this complete folution of the particular cafe may be 
collected in general, that if the odlant ABC be taken fuch 
upon the moving fpher’eal furface, that the track of O there- 
upon may crofs the two great circles AB and AC at right 
angles, a point which is at reft with refpect to both motions, 
or round which they are performed like a fingle motion, may 
at the inftant of the momentary pole’s crofting each of thofe 
great 
