Solution of a Problem in Fluxions. 69 



there are co-ordinates ; but the indeterrainates are to be eliminated, 

 which being done, the number of equations will be less than before 

 by as many as there are indeterminates ; but the equations of condi- 

 tion being the same in number as the indeterminates, there will be 

 as many equations as there are co-ordinates, whence the place of 

 each particle can be found as stated above. This process is virtually 

 the same as to suppose that the effects of the connexions, &;c. of m, 

 /in, &ic. in altering their motions are included in (B^), (B''), &c. 

 among the terms F(^r, Y'tir', fee. and then to eliminate the indeter- 

 minate forces. Another method consists in expressing x, y, z, ,x, 

 &;c. in terms of other variables, which either wholly or in part com- 

 prehend the equations of condition ; then by putting the co-efficients 

 of the independent variations thus obtained, each =0, there will re- 

 sult equations sufficient, wiih the equations of condition, to find the 

 place of each particle m, ,m, h,c. at any given time as before. In 

 the case of the motion of a solid, m, ,m, ,/m, he. are to be considered 

 as elements of the body ; hence by supposing their sum, or the quan- 

 tity of matter in the body to be M, m may be expressed by dM ; 

 then by expressing x, y, &c. in terms of other variables, which are 

 the same for all the elements of the solid, integrate relatively to the 

 mass of the body, considering the common variables as constant in 

 the integration ; (the well known properties of the centre of gravity, 

 and the principal axes of a solid, will serve much to facilitate this in- 

 tegration ;) after the integration put the co-efficients of the variations 

 which remain after having eliminated so many as there are equations 

 of condition, each =0, and there will result equations sufficient, with 

 the equations of condition, to find the place of any given particle of 

 the solid at any given time. 



It may be observed that if the forces mF, mF^, &c. ,m,F, /m'^', 

 &ic. which act on m, ,m, Sic. destroy each other's effects, so that 



d^'K d-Y 

 there is no motion in the system ; then -rr^i -7:75 &c. are each =0 ; 



hence by putting (for brevity) mF=f, m¥'—f', &c. jm¥=J, 

 ,m^'=,f', &c. and so on for f^m, f,pn, he. (D) becomes S(/ir 

 ^^'cV'-f- hc.)—0, (F) ; which is the well known formula of statics. 

 (See Mec. Anal. Vol. I, p. 29, art. 2.) But the formula of statics 

 can be otherwise demonstrated by the aid of the principle of the 

 composition and decomposition of forces. For imagine a particle of 

 matter, considered as unity, to be referred to three rectangular axes, 

 X, y, z, whose origin is at any point of the line r, in which the force 

 F that acts on the particle is exerted ; then F decomposed in the di- 



