Spherical Motion, 505 
natural or momentary axis, and the angular velocity of the 
furface round it are varied. 
Here, other things remaining as in the preceding propofmon, 
inftead of the conftant quantities a , b, and c, we have the 
variable ones x, y, and z. Let the variable fines and cofines of 
AO, BO, and CO, be refpeaively expreffed by b and /3, g and 
7 , a’nd d and 5 ; and let t = the time from the commencement 
of the motion ; then it is well known, that the reipective 
accelerating forces along CB, CA, and AB, muft be expreffed by 
*:, i, and % ; and the radius of the fphere being fuppofed = 
i t i 
unity, the angular velocity about the axis whofe pole is 
z=y 9 e'Szzz, 
O = e ~ s/ x z + y -f-z 2, = es/ + y + § , efi — x 9 
# = 40 + fie 9 y ~?y + ye* $e, £T + y 4* = i > + 7r + ^ 
= 0, ( 3 ! +y=i = & +? = 1 r+r~: -£ 
-V, i= p£^+tf+li=^= j f5=^. A "‘>. 
V -x 1 -\-y + z 
by fpherics, as g : 1 :: l ■ *- = f- OBR=f. QA :: (2 : - = 1. OBS 
={. CQ=cof.AQ, tang. AQ = ^ and the fluxion of the arc AQ = 
^ g u t by the foregoing proportion, BO is perpendicu- 
lar to B b the track of the point B; confequently, as f. OBR 
_£ aQ^: f. OBS = cof. A Q :: z : x; therefore the tangent of 
. rv « j a xi~zi _ <Sxd'+av— ^x?g+>_ ag p £ f ore 
AQ= - and AQ =-*—*- e 2 +f 
therefore, whether e be conftant or variable it makes no differ- 
ence in the expreffion for AQ, In like manner it will appear, 
_• yi—xi V&—&Y 1 pc _ z j—y z _ Moreover, as 
that BR aad " b “ 7+i 5 “ / + r- 
rad. 
1 Z /02 1 . 2 
x -\-y P +? 
Vol. LXXX. 
yT + & 
u u u 
» 
