Spherical Motion . 51^ 
\ e x f. OY = the velocity of the moving fpherical furface at Y> 
which is therefore the angular velocity of the furface at Y 
round an axis at reft whofe pole is Z, becaufe ZY^po 0 
which four values obtain, let the point Z be taken at reft in 
abfolute fpace wherefoever it will. Alfo, e x ft OZ is the ve- 
locity with which the furface pafles under Z in a direction 
perpendicular to the great circle OZ at Z, which mu ft there ^ 
fore be the real velocity of the furface itfelf there at that in- 
ftant; therefore the fluxion of the track upon the furface 
which continually pafles under Z is = e x L OZ x / = 
\/f. Zi 2 + ft Z^ 2 +f. Zr z From which equation, and the pro- 
perties of O found in the preceding propofitions, general ex« 
prefiions for the relation of Z and O may be obtained. But, 
feeing that there is fuch a latitude in determining or fixing 
upon a proper point Z out of an infinity of points at reft, and 
this handled in a general manner will run into a complex cal- 
culus ; in order to fix upon a point Z under the moft eligible 
conditions, it may be beft to deduce them from the properties 
of any particular problem that comes under confederation. 
For example, taking that in the fecond fchollum to the laft 
propofition, where # and e are conftant, and y + — x is 
alfo conftant and = e l y + e'T — e 7 — e 7 (¥ = off, or y 4- T = b 7 alfo 
conftant ; and the velocity with which O fluffs its place along its 
in 
proper track ■= Lz , conftant alfo. Here, in order to fix upon 
a proper point Z, fuppofe the motion to begin when O (fig. 5.) 
is upon the great circle AB at E, and after fome determinate 
time = t, fuppofe the oftant ABC to have arrived in the pofi- 
tion A / B / C / , and that in this time the point O has ftiifted its 
place from E to O, that is, fuppofing the oiftant ABC to be at 
reft in abfolute fpace, while A'B'C - ’ is in motion, on A'B 
Vol. LXXX. X x x taking 
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