Solution of a Problem in Fluxions. 283 



6p = -b, (1), 



4q=-c, (2), 



9p2_36r=^, (3), 



12pg+72s=e, (4), 



1 1 



x^ <' —px^ -\--{p^ -\-Br)x^ -^{2QSf+q^ -\-9pr)x^ - 



1 



-(r2 + 2^s)^2_^2_o, (5). 



These five equations, joined to (B')=0, are sufficient to de- 

 termine y. 



We evidently have, at the same time, 

 1 1 



g (3p3/2 ^4qy)x^J^^{sy^+3psy)x^ =0 ; 



or better, 

 {3py-{-4:q)x'^ -\-2sy^ -\-6ps=0, an indentical equation. 



Art. XII. — Solution of a Problem in Fluxions ; by Prof. 

 Theodore Strong. 



TO PROFESSOR SILLIMAW. 



New Brunswick, June 8, 1829. 



Dear Sir — Should you consider the following solution of 

 a well known problem of sufficient importance, you will 

 oblige me by giving it a place in the Journal. 



Problem. — Supposing that a particle of matter, projected 

 from a given point, in a given direction, with a given veloci- 

 ty, is deflected from its rectilineal course into a curve line ; 

 It is required to determine the equations of its motion. 



Solution. — Let its motion be defined by the three rec- 

 tangular axes (.T, y, z,) x=r. cos. 9 cos. v, y=r. cos. & sin. t?, 



, y X 



t=r. sm. ^, .-'. r^ =x^ -{-y^ -\-z^ (1), -=tan. v (2), -=cos. u 



y 

 cot. 9 (3), -=sm. t^ cot. ^ (4). Let t denote the time, (or 



the independent quantity, which varies as the time, increas- 

 ing by equal elements dt, in equal elements of the time.) 



The question requires that x, y, 2, r, ^, u, be found in 

 terms of «, and constants, (which are to be determined from 

 the initial circumstances of the motion,) in other words, x, 



«/, &c. are to be considered as functions of t. Put X = — -57^ » 



