68 Solution of a Problem in Fluxions. 



small that all its parts maybe considered as having the same motion. 



Hence (B) becomes ml j-z — +For-f F^f^r'-f- &c. 



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, ^priyhc. ; I shall have for ,m, ^^m, ho. 

 -equations of the same form as (B^)* Thus, supposing that ,x ^y ^t 

 are the rectangular co-ordinates of ^m, which are respectively parallel 

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

 -yesponding to FSr, F(>r^y &lc. are denoted by /F^.r, ,V^oy, he; I 



shall have for ,w the equation ym\ —-rp T/Fo/, 



+;F'6y + &:c.)=^05 (B'^). In like manner the formula for ,/m may 

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

 &:c. Now since m, ,m, ,,7/13 &lc. move as a system, or in connexion; 

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

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



formed, I have Sm( ^^ +FSri-F'5r'^hc.j =0, 



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

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



Sm\^ —jf, — +X-^x+Y(5y+Z'^z j --0,(E); the large 



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 ^my and so on for ^/7?i, /^^w, fee. (E) agrees with (P), (given 

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

 equations of connexion between m, ^m, ^^m, &:c. and of the lines, or 

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

 from (E) so many of the variations o'x, (h, oz, 6^x, S^y^ h^%^ rj^'^x, &:c. 

 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 togetlier with the equations of condi- 

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

 ^x, ^Y, fee. ; 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, Stc. 

 becomes known at the same lime. 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 ox, f5Y, 6z, &tc. being considered as independent, their co-effi- 

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



