Solution of a Problem in Fluxions. 285 
these by cos. v, and that of the second by sin. v, then add the 
products, and by (d) I have d (Gr) = —— as 
dt 
(C). The equations (A), (B), (C), are those which I pur- 
posed to find. e solution of the question is now reduced 
to the integral calculus, and the integrals of (A), (B), (C), 
z er a*y @% 
manifestly depend on the values of — ade? aa Ge 
their equals, X, Y, Z, which are involved in F, F’, F”, as 
2 y 
given by (6), (c), (d), respectively. The quantities — “aa 
d?y d? 
~ dt? dt 
ment of the motion, or at some determinate point of the 
described curve, and to vary according to some given law; 
which means their general forms of expression (or X, Y, 
Z) become known. In the language of dynamics, X, Y, Z, 
tap gh d?z 
z *. 
3? are supposed to be given at the commence- 
are the forces which cause the changes —F> — qm ae 
np oGbirdy Gt 35 i= 
in the velocities of the particle 7 4 a’ in the direction of 
be given; that is, if the unit is (1) second, t denotes sec- 
onds.) Also, F, as given by (6) is the expression which 
would result by decomposing X, Y, Z, in the ares of r, 
v 3 
the usua! rules of decomposing forces, and F’, F’”’, res 
