OF THE MOTION OF A SVSTEM. 225 



other bodies, considered as at rest, on any 

 given plane, multiplied by the products of the 

 two masses respectively, is equal to the pro- 

 duct of the mass of the whole system, into the 

 sum of the projections of the areas described 

 by the separate revolving radii round the com- 

 mon centre of gravity on the same plane, mul- 

 tiplied by the separate masses respectively. 



Since c-=z'Em — --r— — » ^^^ obtain, by proper substitu- 



^ tions elm— 1mm'. ^—^^ li-ll — ILI \ 



dt 



! [For c^m—(m + ?» + m". . .) (m — '-^-—^ 1- m — -^ — 



d^ d^ 



+ 4 . .) in which all the binary combinations of the different 



values ofm occur once witli each of the two corresponding 



fractions ; and taking the first two for example, we have 



. , ^. / xdy — yd.r , x'dy—ydx\ , , , 



{m-\-m). \rn, —^—r^ — 4- m'. — ^-—y- j = {m7n + mm')xdy 



— {mm + mm')ydx + {mm' + m'm')x'dy — {mm' + m'm') y'dx'y 

 neglecting the divisor: but »m' {x' — x) {dy'—dif) — 

 ^y—y) (d^'— da-) = mm' (x'dy — x'dy—xdi/ + xdi/ — y'dx' 

 A-y'dx + ydx'^/djc), the difference of the two expressions 

 being mmxdy—mmydx-\-m'mfx'di/' — m'm'y'dx' -\- mm' x'dy — 

 mm'y'dx + mm'xdy — mmydx' , or m{mjc + m'x')dy—m{my + 

 nty') dx + HI {mx + mx) dy'—m'{ rny + my)dx' : and if we 

 added together any others of the bodies in pairs, it is ob- 

 vious that the coefficients of the fluxions in this difference 

 would make the series mx + mV-f mV + . . . ^Swj^'zzO, 

 by the property of the centre of gravity ; consequently 



Q 



