Spherical Motion* 5 1 ^ 
furface which at firft was under Z will arrive at s in Aft? 7 ’ 
where A's=AZ, and E confidered as the fame invariable 
point of the furface will arrive at e, fo that A^izAE; there* 
fore, fince EZ = OZ is conftant, and Z at reft both with 
with refpedt to the velocity eb of A / round it, and the velocity 
— with which O fliifts its place, itmuft be as f. EZ — f. OZ : 
f. AZ :: ^ : eb :: £ : 1, but b and /3 are the fine and cofine of 
AE = A'Q— EZ + AZ ; therefore, as f. EZ = b x cof. AZ — /3 
x f. AZ : |S x f. AZ :: I : r, and as b x cof. AZ : @ x f. AZ :: 
A 
I + r : 1 :: 4 - — tang* AE * tang* AZ 2= | x — - 
A 0 ° & £ A-f 1 
, and f. AZ : 
cof. AZ — f. BZ :: : A-f x x /3 :: tang. AE * f. AZ 
A 
A£ 
— 9 and cof. AZ = -£===. ; and thus a dif- 
^A a + *A|S»-i-jS* + 
tin£t idea of this motion of the fpherical furface is obtained, it 
being now clear, that the point A / moves round Z at reft 
with the velocity eb , and as f. ZA : 1 : t eb : 
e i SA* + 2 A0 z ±I3 & 
A 
the angular velocity with which A / moves round the axis 
whole pole is Z, which is therefore conftant * and at the fame 
time the furface itfelf moves in the direction of the great circle 
B C 7 , that is about the axis whofe pole is A 7 with the conftant 
velocity x — e/ 3 , which two motions may be confidered as fepa- 
rate, and the reft as confequences of them ; that is, the point 
Z is at reft, and the point A 7 moves uniformly round it, whilft 
the furface upon which A / is an invariable point moves round 
the axis whofe pole is A 7 with an uniform angular velo- 
city, thefe two angular velocities being in the ratio of 
X x x 2 K / 
