536 NOTE ON A PAIR OF DIFFERENTIAL EQUATIONS IN THE LUNAR THEORY. [482 
in the equations the coefficients j, k [regarded as each of them = &]: in fact, the 
developments could then be arranged according to the powers of k, that is according to 
the powers of the disturbing force; whereas, when k is taken =1, we have only a 
development in powers of m, and since m also presents itself through the coefficient 
2 — 2m of t in 2v — 2mt, terms which are really of different orders in regard to the 
disturbing force, are united together into a single term: so that, instead of a term 
of the form (Ak + Bkr + &c.) m p , where A, B, are numerical, we have the term 
(.A + B +..) mP, where of course A + B.. is given as a single numerical coefficient. 
There is no equal advantage in retaining the two coefficients k, j, as this only serves 
to show how a term arises from the central and tangential forces respectively; thus 
retaining these coefficients, the integrals as far as m 2 are 
v = t + m 2 sin 2D, 
- = 1 + £ m 2 k + {^k + \j) m 2 cos 21), 
P 
agreeing with the former result when k =j — 1; but there is, nevertheless, some interest 
in retaining the two coefficients. I hope to develope the results somewhat further, 
and to communicate them to the Society.