68 Solution of a Problem in Fluxions. 



small that all its parts may be considered as having the same raotix)n. 



Hence (B) becomes ml -j-; \-F5r-{-F^5r'-\- he. j 



=0, (B'), for the motion of m; for (B) is evidently to be taken as 

 often as the unit of particles is contained in m. Now supposing that 

 the moving particles are m, ,m, y,m, he. ; I shall have for ,7n, „m, &c. 

 equations of the same form as (B'). Thus, supposing that ^x ,r ,z 

 are the rectangular co-ordinates of ,m, v\?hich are respectively parallel 

 to X Y z and have the same origin ; and that the quantities cor- 

 responding to Fi)>, F<3r', &:c. are denoted by ,Fd^r, /F^liy, &ic. ; I 



/ d" ,xf)y^-^rd" ^YS^Y-[-d'- ,zS,z 

 shall have for ,in the equation pii ^7^ T/Fo,?-, 



~\-,F'6,r'-{- &£c.) = 0, (B'^). In like manner the formula for y^m may 



be denoted by writing two marks below the letters, and so on for ,,,171, 



&£c. Now since m, ,m, „m, &C. move as a system, or in connexion 5 



it is evident that the equations (B'), (B^^), &;c. must be added; hence 



supposing (for brevity) that S written before (B') denotes the sum thus 



/d'X'^-s.-{-d'^Y()Y+d-z5z , ^ ^ ^ N 



formed, I have Sm( -^, +F(5r+F'oV-f Stcj =0, 



(D); which is the general formula of dynamics. (See tlie Mec. 

 Anal, of La Grange, Vol. I, page 251.) (D) can be changed to 



S„(^!^^^i^'±f^Vxix+Y.V+Z!.) =0, (E); Ihelarge 



capitals X, Y, Z, denoting the same things as in (C), (given at page 

 333 of the last Journal,) ^X, /Y, ,Z, being the corresponding quanti- 

 ties for ,m, and so on for ,,m, ,,im^ hz. (E) agrees v^ath (P), (given 

 by La Place in Vol. I, p. 51 of the Mec. Cel.) By means of the 

 equations of connexion between m, ,m^ ^/in, he. and of the lines, or 

 surfaces, on which they are supposed to move ; we are to eliminate 

 from (E) so many of the variations 5x, (5y, H, (5,x, 8;s:, o,z, S^^x, &lc. 

 as there are equations ; then since the remaining variations are inde- 

 pendent of each other, their co-efficients must each be put =0 ; and 

 there will arise equations which together with the equations of condi- 

 tion will make as many equations as there are co-ordinates, x, y, z, 

 ,x, ,Y, &:c. ; by which each of the co-ordinates can be found at any 

 given time, and hence the place of each of the particles m, ,m, Ssc. 

 becomes known at the same time. But the same tiling can generally 

 be more expeditiously effected by adding to (E) the variations of the 

 equations of condition, each multiplied by a separate indeterminate ; 

 then (5x, OY, (5z, &cc. being considered as independent, their co-efS- . 

 cients must each be put —0; which will give as many equations as 



