Spherical Motion . 525 
very equations brought out by Mr. Landen in fo very different 
a manner. 
Here then the matter may be fafely refted ; for the accelera- 
tions are moft certainly as the accelerative forces, and not as the 
motive ones. Conclufions, therefore, that are drawn from a 
contrary fuppofition cannot be true. 
It may not, however, be improper to (hew here how Mr. 
La n den’s motive forces E and E /7 arife from thofe above 
brought out; thus, in fig. 3. Prop. iv. let s and t the fine and 
cofine of AQ to radius 1, that is, let s 
motive force along BA refolved into the direction BO becomes 
— x c A - X e*( 3 yt 9 and that along BG refolved into the fame 
diredion BO becomes — x a — c X cyh^ the difference of thefe 
3 1 
— and /= then the 
g g 
ss — x e z y x d 2 — c 2 x $s - c 2 - b L x $i = 
3 3 
X O/ — C muft be 
the motive force ading along the great circle BO in the fenfe 
from B towards O, or from O towards Q; and this is the very 
motive force E determined by Mr. Landen, and ading in the 
fame manner. The motive force which ads at O perpendicu- 
larly to the force E is moft readily obtained from that ading 
along CA ; for if a tangent be drawn to the great circle BOQ 
at O (fig. 3.) it will interfed a radius of the fphere drawn 
through Q at a diftance ( - j from I the centre of the fphere == 
the fecant of the arc OQ, and as 1 : - = that fecant :: the force 
g 
MD//3J 1 Tn r i ^ MV e 7, git 
ading at the diftance QJrom the centre : — — ~ - 
the force adting in the plane of the great circle CIA at the 
diftance - from the centre I, and perpendicular to a tangent at 
' O 
