PreceJJton of the Equinoxes. 519 
freely in thofe directions; and let av, b r, c c, reprefent 
the accelerative forces of the refpective bodies, as altered 
by their mutual aCtions upon each other: then, becaufe 
c x c c is the moving force gained by c, and axw + bx r r 
the moving force loft by a and b, regard being had to 
the lengths of the different levers af, be, we ftiall have 
a x w x AE + BxtRxBE equal to cxccxce, that is, 
AxAExAv-At)+BxBExBR-Br equal to cxctxCE, and 
by tranfpofition ax aexav + bxbexbr equal to cxcex 
cc+ax aex Ar+BxBExBr. Let s, i, reprefent, as in 
art. 2. the fpaces which would be defcribed by the bodies 
A and b at liberty in any very final! portion of time, and 
let x be the fpace which a actually defcribes in that time 
when connected with b and c by the lever ae. The 
quantities will then be the fpaces defcribed 
byB and c refpeCtively ; and, laftly, becaufe the fpaces de- 
fcribed in given times are as the accelerating forces, the 
above equation gives x equal to 
A X AE X J + B X BE X J X AE 
axae+bxbe 2 + c xce** 
The fame method extends itfelf eafily to more diffi- 
cult cafes, and by its affiftance feveral very important 
theorems are briefly demon ftrated. 
§ 14. The reafoning made ufe of in art. 6. will ap- 
pear very evident to any one moderately verfed in the 
Y y y 2 elements 
