Solution of a Problem in Fluxions. 333 



&tc.=0(A) ; since Sx, Sx', he. are evidently equal to each other, as 

 also Sy=Sy'=:Sy'^ = hc. the same may be said of Sz, ^z', h,c. I now 

 suppose the motion of the particle to be referred to three rectangu- 

 lar axes having their origin at any given point in space, these co-or- 

 dinates which I shall denote by the small capitals x, y, z, being re- 



spectively parallel to x, y, z', then -jji = the accelera- 



tZ-x d^y-\-d^y'-[-%ic. 

 tion in the directionof x may be denoted by i and -j- = 



fZ^Y d^z+d"z'-^hc. d'-z , , , , , 



^W Jt~' '^~dF'' ^x=da?, 5Y=6y, 5z=8z', by 



<Z-x^x-{-^"Y^y4-^^z<^z 

 substituting these vahies (A) becomes -rr^ -f- 



-|-F(Jr+F'(5r''+&ic. = 0(B). I have supposed F to tend to dimin- 

 ish r, Y' to diminish r', Sic. but should any force act from its origin 

 or tend to increase its distance from its origin, the variation of its dis- 

 tance must have the sign minus in (B), thus if F'' tends to increase 

 r" instead of -{-Y"8r" I shall have —Y"8r" in the formula. The 

 formula (B) is the general formula of Dynamics in the case of the mo- 

 tion of one particle of matter. (See the Mec. Anal, of La Grange, 



/ d-x \ 

 vol. I. page 251.) (B) can be changed to I -^t^+X 3 ^x-{- 



(d-^Y \ /d-z \ . . „ 



>--+Y I^Y+ {-^--\-Z)Sz = (C) ;bytaking the variation of r, 



expressed in terms of a?, y, z, and o( r'='^x'^ -{-y'^ -{-z'^ &;c. then 



xF x'F' y¥ y'¥' 



puttmg the large capitals X=— -+-^+&c.Y=— +-^+ &;c. 



zF z'Y' 

 Z=— 4- — 7-&1C. The formula (C) agrees with (/) given by La 



Place, (Mec. Cel. vol. 1, page 21.) If the particle is free, the 

 coefficients of ^x, Sy, Sz, must each =0, which gives -^77- +X=0, 



d-Y d'z 



^—-{-Y=0, -j---{-Z=0. But if the particle is supposed to move 



on any given line or surface, then by means of the equations of the 

 line or surface, we are to eliminate so many of the variations Sx, <^y, 

 ^z, as there are equations, and to put each of the coefficients of the 



