12 The Rev. Brice Bronwin on the 



Tvr d *P , ,d*y-d*y' „dll 



M m ^T% + m m ,, 2 — — = N -7—, 



dt 2 at 2 dy 



Tvr ,d*y' ,d*y-d*y' XT dR 



Mm -jw — mm' — V—r-s — y - = N- r - 7 , 

 dt 2 dt 2 dy 



,, d*z ,d*z-d*z' „dll 



M m To. + mm' -j-z = N -^— , 



dt 2 dt 2 dz 



,, ,d*z' ,d*z-d*z' XT tfR 



M m tj — mm' -j-s = N -r-.. 



dr dt 2 dz 1 



Whence we easily obtain 



,, dx 2 + dy* + dz* , T . da!* + dy 12 + dz 12 

 Mm rf? +Mm tfc 



,dx"* + dy m + dz" 2 „ XTO . 



+ m ml * 2NR = i, 



dt 2 



the result being integrated, and x" put for #' — x, &c. 



If we eliminate dx, dy, dz, dal, &c. by their values in 11, 

 v, and w; 11', t/, «/, as in (F.), we shall have 



M/i „^±^±*f + iw «w±f«!_ 2NR=i . ( M. ) 



Thus the square of the velocity of the centre of gravity of m and 

 m! about M, multiplied by M {m + m'), added to the square of 



the velocity of m! about m, multiplied by m m! - t — , is 



equal to 2 N R plus a constant. Two other similar formulas 

 may be found relative to the centres of gravity of M and m, 

 and of M and m! 



By the nature of the function R, we have 



dR dR ,dR ,dR 



x -j— — y -5 1- x' -r—. — y' -=— 1 = 0, 



dy y dx dy 1 * dx 1 ' 



dR dR , dR ,c?R_ 



dz dx dz' dx 1 -'■ ' 



dR dR ,dR ,dR '* 



* dz dy u dz' dy' 



By the aid of these properties we may easily derive from (L.), 

 or the equations following them, the equations (E.) relative to 

 the areas described. 

 We find from (A.), 



,, xd*x + yd 2 y + zd?z , ,, ,x' d % x' + y' d 2 y' + z' d*z' 



M m y ,,g + M ml ,,a 



dt 2 dt 2 



