Solutio7i of a Problem in Fluxions. 285 



these by cos. r, and that of the second by sin. r, then add the 



products, and by (a) 1 have di \ + =r 



\ dt } dt^ 



IT 

 (C). The equations (A), (B), (C), are those which I pur- 

 posed to find. The solution of the question is now reduced 

 to the integral calculus, and the integrals of (A), (B), (C), 



d^x d^y d^z 

 manifestly depend on the values of — jTJ' "^^Ta' "^T/I' ^^ 



their equals,- X, Y, Z, which are involved in F, F', F", as 



given by (6), (c), (a), respectively. The quantities — -j-^^ 



d^il dr^z 

 — -jT^i — -,7^' are supposed to be given at the commence- 

 ment of the motion, or at some determinate point of the 

 described curve, and to vary according to some given law ; 

 by which means their general forms of expression (or X, Y, 

 Z) become known. In the language of dynamics, X, Y, Z, 



d'^x d^y d^z 

 are the forces which cause the changes — "Xl' - "j7i' ~~J^'' 



dx dy dz 

 in the velocities of the particle -j ' -5:' -t-> in the direction of 



the axes x^ y^ z, respectively in an assumed unit of time, (as 

 (1) second for instance in terms of which t is supposed to 

 be given ; that is, if the unit is (1) second, i denotes sec- 

 onds.) Also, F, as given by (6) is the expression which 

 would result by decomposing X, Y, Z, in the direction of r, 

 by the usual rules of decomposing forces, and F', F", res- 

 pectively denote the changes caused by the action of the 

 forces (in the assumed unit of time) in the areas described 

 by the orthographic projection of r on the plane of (x, y), 

 and by the motion of r in a direction perpendicular to the 

 plane of (.r, y) ; in other words, F' is the moment of the for- 

 ces which act upon the particle decomposed in the direction 

 of the plane {x, y), and F" is their moment decomposed in 

 the direction of a plane passing through r, at right angles to 

 the plane {x, y). The equations (A), (B), (C), are the same 

 as the equations (H); given by Laplace at page 149 of the 

 Mec. Cel. they become much simplified when the conditions 

 of the question are such that the motion of the particle is 

 always in the same plane ; for supposing (.r, ?/) to denote 

 the plane, I have c=0, .'. <9=0, and the equation (C) does 



