Solution of a Problem in Flucions. 333 
d?y+d?y + &e. d?z+d? z/+&e. 
(aster tte), (Babee) einai 
&e.=0(A) ; since dx, dx’, &c. are evidently equal to each other, as 
also 6y=dy/ =dy’=®&c. the same may be said of dz, 42’, &c. Inow 
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, ¥, z, being re- 
: d?x+d?7/+&c. 
spectively parallel to w, y, z; then We = the accelera- 
Rae d*x d?y+d7y+e 
tion in the direction of x may be denoted by ang 2 toe Tee 
d?y @zt+d?z/+&ec. a? pod 
mae? i 3 Berd a? and dx=dx, dv=dy, da=dz; by 
t 
d?x0x +d? yvdy+-d?2zdz 
dt? + 
substituting these values (A) becomes 
+Fér+F’ir’+&c.=0(B). Ihave supposed F to tend to dimin- 
ish r, F’ to diminish r’, &c. 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 merease 
r’ instead of --F”ér’ I shall have —F”ér” 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, 
es 
vol. I. page 251.) (B) can be changed to be +X) Ox-+. 
+ (2 +Y) iy-++ ( o" +Z)in=0 (C) sby taking the variation of r, 
expressed in terms of 2, Y, 25 and of 7’ =V zt ty 2127/2 &e. then 
ak. a’ }Y yf. Pee 
putting the large capitals Ka 4+ eI 7 &: 
aa i The formula (C) agrees with (f) given by La 
Place, (Mec. Cel. vol. 1, page 21.) If the series free, the 
Coefficients of dx, dy, dz, must each =0, which gives Fa +X=0, 
d? 2 : E hvickets 
iz +Y=0, = +7=0. But if the particle is suppers a 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 62, oy, 
éz, as there are equations, and to put each of the coefficients of the 
