10 The Rev. Brice Bronwin on the 



With equal ease we find from (B.), integrating results, 



,, xdy—ydx ,, .x'dy'—y'dx' ,x"dy"—y"dx" 



Mm — 2__Z — -j- Mm' — Z—rf (- mm' ^— jf = c, 



dt dt dt 



M 



m 



xdz—zdx 

 ~dt 



+ Mm' 



x'dz'—z'dx' 

 ~di 



-f mm 



,x"dz"-z"dx" 



— J 



dt 



HI 



_. ydz—zdy , A , .y'dz'—z'dy' .y"dz"—z"dy" ., 



M;»- — 3- — - + Mffl'^ tt — — + mm'Z — ^- = c". 



dt dt dt 



These last are known. 



,, , mx + m'x' my + m'y' mz+m'z' 



Make ■ 7- = «, — — • — r- = v, ■ — j- m to ; 



m + m m + m' m + m! 



u, v, and <oo will be the coordinates of the centre of gravity of 



m and m'. To simplify, let x" m x' — x = «', y" = y' — y = tf 9 



, „ . a" ?/ — m' u' . a" u + mu' 

 z" = z' — s = w; we find # = r . , v = » , 



JW," ft," 



&c. Substituting the values of x, y, and #, x 1 , y\ and «', and 

 their differentials in (E.), there results 



», „udv — vdu Nfflffl 1 u' dtf — t/ du' 

 M/x" ^-t— - + » .. 



Mix" 

 Mfi" 



dt /*" d* 



udiso—wdu T$ m m! u' d wf —id d u' 



dt 



+ 

 + 



dt ' p." dt 



vdw—tsodv , Nmm' xl dw' —iv' dt/ 



fir 



d* 





Thus the area described by the centre of gravity of m and 

 m! about M, multiplied by M (m + m'), added to the area de- 

 scribed by m' about m, multiplied by (M + m + rn') 



m m 



7« + m' ' 



on each of the coordinate planes is a constant quantity. Two 



other similar sets of equations exist relative to the centre of 



gravity of M and m, and of M and m'. 



„, ,." , xdy — ydx xdz — zdx 



To abridge, we make ^ff = c v j} = c 2> 



ydz-zdy _ x'dy' -y'dx' , 



IT — ~ C3 ' — ~dt~ - c i' &c " 



x"dy" -y"dx" _ „ 



dt 



= c" v &c. 



The following are identical : — 



h * — c * y + c i * = °> ^s d - ^y + ^i *' = °» 



c" 3 .r" - c\y" + c" x *" = 0, .*• de 3 - y dc % + * c^ =0' ^ (G.) 

 a?d(? a -y'd<! i + z'd(? l =0 i x"dc" 3 -y"dc"^z"dc\ = 0' 



