DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
307 
From these expressions the values of {«, e}, {a, /}, &c. are found with the greatest 
simplicity, and the results are 
mf^{a, (a)}=2wa^ e}- 
na 
m^{w,e) = , 7n/^{(a), tan-, 
r 1 . i , , na 
the rest all vanishing. Hence, observing that if R be taken in its usual signification 
we have n = — R, we obtain* 
#) 
j«/e' =- 
■na 
r 
ybw' =na<, — 7 ^==tan 7 ; 
de Vl — i 
2 di j’ 
I — f 1 <^R , . </<^R , ^/R^'l 
- vr^Aim, 2\W)'^'di)y 
na 
fijV = — 
dR 
dR 
sim -v/ 1 — dt 
in which we may, as usual, put a for (a), provided that in forming the term nt be 
exempt from differentiation with respect to a. 
56. A comparison of the above process with that by which the corresponding 
* If we consider R as a function of p, q instead of /, v, where ^=tan < cos v, 2 = tan i sin v, we find 
dR dR> 
di 
and consequently 
dv 
,( dR . dR\ 
= sec®j ^cos y + sin v j 
t . / dR . dR\ 
V dq dp)’ 
, — na(secj)®r dR , , i /dR , dR\'| 
-v/i_e2 dq^ 2 Vd(e) daryj 
,_wa(seci)®f dR i /dR dRNl 
The formulae will then agree with those of the Mdcanique Cdleste (Supplement to vol. iii. p. 360, ed. 1844), if 
we allow for the dififerent mode of measuring longitudes, and neglect, as Laplace does, terms of the second 
fiR 
order with respect to t and — . (Laplace uses R with the opposite sign.) Those in the text agree (allowing 
for notation) with the expressions given by Professor Hansen, Astr. Nachr. No. 166, art, 3, equations (2). 
mdccclv. 2 T 
