70 Central Forces. 
i d*Q’ 
x=0 in (6), I have to da for the general form of the 
d" Q’ d’Q 
: oe - the en ee 
quantities Q,,Q,...Q, sto 755 aaa sxthe valueol 53. aie 
/ 
dQ 
when »=05) let n=1, 2, 3, 4, and so on successively, then Q, =a 
2 / d 
Q,= ee Q,= 12. 2 and so on; by substituting these values 
dQ’ x? d?Q’ 
of Q,,Q,, &c. in (a) it becomes Q= QV+a75 tie te +353 ~ 5 
= etc. (c). Let r=fy= any function of y; and y=1(u+ar) 
(d), =any function of u+-ar; then if (d) is solved with respect toy, — 
y will equal a function of u and «, but as there is no given relation — 
between wu and a they are independent variables; also Q=Fy=a _ 
function of wanda. Again, (by solving (d) with respect to u-+ar;) — 
' w-+-ar=a function of y, let z denote this function; then (d) becomes 
dz 
z=u+ar ie The differential of (e) with respect to u, gives dy x 
2 di dz dy _ 
alte qi,’ 2nd its differential relatively to x, gives 7 dy Fe 
ye dQ dQ d 
cae. hence 7 = a = 2 (fs and qe dy xo} but, by (f)s 
dQ dy dQ _dy dQ ie ae Q “dQ AQ 
dy *da="* dy “du [since Geta) rages hence Ge =r 
(g). Let R= any function of r; then R=a function of y =a fune- 
—_>_ 
tion of u and-x; hence ae gil dy * aa di +R.d =qy 
a(R) dR dy dQ Ge) 
dy 
dx 
xgeXgeX nal n)= a(R a) Now by (g) I have 
dQ re 
wo Min) ri, i butby(g) ears « ; aaa 
ara) ge d(T) ele zs) 
by (h); or, by (@)s 
dx? ~ dz.du-—~— du” 
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