332 Solution of a Problem in Fluxions. 



parabola, which gives V : V'': '.\/2 I 1 (11) ; (P. prop. 16. cor. 7.) 

 Since (10) and (11) are independent of the parameter of the conic 

 section, by supposing the parameter to vanish the sections become 

 right lines, and the particle describes a right line, as stated by New- 



, ^ /AB 



ton, B. 1. sec. 7. propositions 33 -. 34, andV : V^• : VAC : V -^ 



as in prop. 33, for in his fig. (1) 2a — r=AC, in fig. (2)2a-|-?'=AC, 



AB . 



in both a=-^ ; but in prop. 34 V : V' : ! ^2 I 1. 



I will conclude this paper with some remarks on (i) which pre- 

 sented itself in the differentiation of (1) of the Journal (for July). 



(d-x xF\ /d'y yF \ 

 It may be changed to \^^+T/^+ \^+"7 /^+ 



-^1 -177-4- — jz=0 (a). If the particle is free, that is, if it is not 



supposed to move on any given line, or surface ; then x, y, z, are in- 

 dependent of each other ; and their coefficients, or the quantities in 

 (a) within the parentheses must each =0, in order that (a) may be 



d'-x xF d~y yF d'z zF 

 indefinitely trne. Hence ^-= "— ' ^= -^-' W = -~'' 



which are the equations {g) assumed at page 69, of the last Journal, 



from the ordinary methods of decomposing forces. Again (6) can 



xd^z-\-yd-y-\-zd"z 

 be changed to jr^ -{-Fr — (c) ; similarly I shall have 



x'd^x'+y'd'y'+z'd^'z' 



^777 --fFV— (c') should the particle be acted on 



by any other force, F', analagous to F; x', y', z', being respectively 

 parallel to x, y, z ; their origin being at the centre of F' ; also r' is 

 the distance of the centre of F^ to the particle. Also I shall have 

 for another force F'', an equation of the same form by accenting 

 X, y, &;c. twice ; Stc. for F'^^ to any number of forces whatever. 

 In order to obtain the combined effect of F, F', F'', &,c. on the par- 

 ticle, I take the sum of the equations (c) &ic. then I suppose the po- 

 sition of the particle to be varied infinitely litde, but so as not to af~ 



. „ _, „ d-x d^x" d^y d y' d-z d^z' 



feet F, F^ ^c.-^.-^ &c. ; ^-, -j^ he. ; ^, -^ he. ; in oth- 



er words, I regard these quantities as constant in taking the variation of 



(d^ X -\- d- x' -^hc\ 

 —- ' j ^x-\- 



