188 CELESTIAL MECHANICS. I. iii. 16. 



The finite integral being understood as comprehending all 

 the combinations of the different bodies in pairs. [Thus for 

 two bodies, m and wi', Xm being m-\-m', Xmx:=imx + m'x\ 

 and "2tmm' :=.7nm\ vfQ\\?L\Q{l,mxy—m^x'^-\-vi'"x^-\-2mm'xx'zz 

 l.mx", 2m — mm\x' — xf ■=. {mx^ -f m'x'") (m + m) -- mmXx'^ + x'^ 

 ~2a:'x)=m2x2"H-m»iV2+m'ma;2" -^-m'^x'^ ^'--mm'sf^—mm' 

 x^'^ + 2mm' XX* : and adding a third body, if S/wx be mx 4- 

 mV + ?»'V',we have {^mx) ^=:m''x'^ + m'"x'^-^m"^x" -{■2mm' 

 xx' -\-2mm'xx" + 2m'm"xW=z (mx-^m'x'- + m!'x"^) {m + m' 

 ^m")^mm' ix'—xf—mm" {x"—xf—m'n^' {x'—xy-, and a 

 similar proof may be extended to any number of bodies.] 



By this mode of computation, we may determine the 

 distance of the centre of gravity from any fixed point, when 

 we know the distances of the different bodies of the system 

 from this point and from each other : and when the distance 

 of the centre of gravity from any three points is thus 

 found, its situation is in all respects completely ascertained. 



Scholium 2. The denomination of " centre of gra- 

 vity" has [sometimes] been extended to any system of bodies 

 with or without weight, as determined by the three coor- 

 dinates X, Y, and Z, thus computed [,but it is more 

 correct to employ, in this sense, the term " centre of 

 inertia" (298)]. 



§ 16. Conditions of the equilibrium of a solid of any 

 figure whatever, P. 46. 



309. Theorem. For a single solid body, 

 whatever its figure may be, we have the same 

 conditions of equihbrium as for a system of 

 bodies^ substituting fluxions nnd fluents for 

 single bodies and finite integrals : that is 



