Spherical Motion . 
53 1 
muft now become M a 2 , M£ 2 , Mr, M x o L - c\ M xr-/, 
M x « 2 - b~. 
a 
2 A 
“ 5 — 
— , and 
2 7 2 
b 
And the three funda- 
mental equations for the accelerative forces become 
■c x yz 
a 
— 5 
t 
c a X *x y 
b‘ 
and 
a L — b X xy z 
or x = 
0 — c X yzt 
a 
» 7 
a" X zxt 
Z 
a —b* X xyt * 
; multiplying the fir ft of thefe equa- 
tions by a z x, the fecond by by, and the third by c 2 z , and 
adding all the three products or refulting equations together 
gives a 2 xx + byy -f- c 2 zz = o ; alfo multiplying them refpec- 
tively by a*x 9 by, and c A z, and adding the three produdts pro- 
duces cfxx-ytfyyy c*zz = o ; and if 2, SB, and C? be the re- 
fpedtive values of x, y, and z 9 at the commencement of the 
motion, taking the fluents a 2 x z + b 2 y 2 -f c 2 z = a 2 + b 2 W + 
and c£x % -f b *y z -j- c z 2 = a 4 Sl z + b fig 2 -f c (££% which therefore are 
conftant quantities. But M a 2 x 2 , Mby % and Mc 2 z 2 , are the 
refpeftive vires vivcz of the body round the three permanent 
axes, and confequently their fum, or the whole vis viva is 
always the fame conftant quantity. Alfo, fince i 
2 • 
a x 
“2 2 . . 
b — c Xyz 
i • 
c z 
b z v 
a z — b 2 Xxy 
fluents 
, therefore 
a , xz 
2 • 
a xx 
b z -S 
b Z yy 
2 
C ZZ 
a 
b z -c z 
X7-3 1 
2 Z 
— a 
a 
, and the 
2 2 
c —a 
xy 2 -W = - — r? x sr-£ 2 ; hence then 
^ a —b 
y 
a X c 
a L 
X x 2 — 4- 315% and z : 
4 
a z X a z — b 
xx z -r.+ 
b z xb z -c % ' c z xb z -c 2 
which values fubftituted for y and z in the equation i 
a x 
~ 9 give i in terms of x , x and conftant quantities. But 
b z —c z Xyz 
the fluent, though attainable by means of the arcs of the 
Z z Z 2 
conic 
