288 Motion of a System of Bodies. 



ordinates of the centre of gravity of the solid; then S{x'^-{-y'^)dm 

 = S{^x' +,y')dmi-2XS,xdm+^'^^^ydrn+{X' +Y')Sdm, but Sdm 

 =m, and by the nature of the centre of gravity S,xdm=0, S,ydm=0, 

 hence S{x'- +y'^)dm=S{,x^ +,y^)dm-\- {X^ +Y')m, {z). 



{z) vi?ill enable us to find the moment of inertia relative to an axis 

 draw^n through any given point when the moment of inertia is known 

 for a parallel axis passing through the centre of gravity of the solid; 

 it is also evident that the absolute minimum moment of inertia be- 

 longs to one of the principal axes which passes through the centre 

 of gravity : see Mec. Cel. Vol. I. pp. 75, 76, etc. 



It has been supposed in 11. that the system revolves about a centre 

 of force situated at the origin of the coordinates, but this is not ne- 

 cessary except for simplicity, for the origin may be taken at any 

 point, (at pleasure,) provided all the forces are considered as disturb- 

 ing forces. 



Hence (11.) have place as before, (there being now supposed no 

 centre of force at the origin,) and the invariable plane is found in the 



D D' . .. . , . 



same manner ; the areas ^j -^j he. being now rectilineal triangles m- 



stead of curvilineal sectors, but this does not affect the determination 

 of the invariable plane. From what has been said, it is manifest that 

 when the system is affected by no foreign forces, there will be a par- 

 ticular invariable plane for each point of space. Again (7) are easi- 



]y changed to Sm[—^^-f j=Sm{Qx-Fy), Sm[ ^f, j 



= Sm{Rx-Fz), Sm( ^ '^"^^ '^ ) =Sm(Ry-Q^), (18), where P, 



Q, R, P', &1C. are supposed to be the. same as in (1), (2), (3) given 



r^dv xdy — ydx 

 at p. 40 ; for as at p. 42, c="~7T-= -iz ' &tc. and by resolving 



Q and P at right angles to r, we have the resultant of all the forces 

 which affect a unit of m when resolved at right angles to the extrem- 



Qa;— Pj/ 



ity ofr= =T, .'.Tr=Qa; — Py, and in a similar way Ty= 



QV — P'!/', and so on ; .'• by substituting these values of c, Tr, c', 

 &;c. in the first of (7) it will be changed to the first of (18), and in a 

 similar manner may the second and third of (18) be obtained from 

 the second and third of (7). (18) can easily be found directly from 

 (1), (2), (3) ; for multiply the first of (2) and (1) by a; and —y, re- 



