Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

76 PLANETARY TIIEORY.
cl.r cldr
O = 2r dt- t dt r - an {\/(1 - e2). 0
dt dt
*+3nfJfJ dR J dR
d ce dr
Or = 1 d (2rêr) 1 dr I
Or (O - - —,e,~)
aun = /(1 - e) dt r dt J
3 r rd R r dR
+ ~ ~ ~ +an/(1 -- e) rt d
+a2(1- de a2/(n - le2)t dr
94. Pitop. 35. To examine the terms of R which are
increased most in integrating the equation for c0.
For the complete calculation of 0, the value of rSr
found in (89) and (90) is necessary. The terms which have
been increased in the integration for rir will not have their
value materially altered in the integration for è0. For, let
cos (pnt - qn't + Q) be one of these; then (92) pn-qn
does not differ much from =6 n: and therefore the first term
in the value of S0 will not be altered by the differentiation.
95. But the terms which are most important iul 0
are those which depend upon terms ill R of the form
P. cos p(nt + e) - q(n't + c') + Q,
where pn - qn' is very small. For, with respect to this
term,
dR
( = p P. sinp(nt + e) - q(n't + e) + Q;
de
2 r rdR
whence I
aw c a ( -e) J Jt de
p 3P
(pn= P-7. q,' e). - )sin P (nt + e) -q't + e') + Q.
(pn - qn')' ^/(l - e~)
Here the coefficient is divided twice by pn - qn': and of
course if that quantity is small, the coefficient is very much
increased in magnitude. The numbers p and q must always be integers: if then two integers p and q can be