53b Mr. Hellins's improved Solution of 
have that convergency, is when n (which answers to his s,) 
is = a case which does not often happen ; however, from 
the values of A and B, when n = — b, he derives their values 
2 
when n — by another very ingenious device, worthy 
of that skill for which he is justly celebrated. But, by the me- 
thod now proposed, the chief part of the convergency is in the 
literal powers ; and such a difference in the numeral coeffi- 
cients, for a different value of n , does not take place. 
For Mars's perturbation of the Earth's motion, the literal 
powers by which the three different series converge, are nearly 
as follows : 
M. Euler's, "j f ^ ; 
M. De la Grange’s, >by the powers of<j 
T he series now proposed, J l T V * 
If, indeed, the perturbation which arises from the action of 
Jupiter upon the earth was to be computed, M. De la Grange's 
series would be the best that has hitherto been published for 
the purpose, as the literal powers of it would, in that case, be 
* For obtaining nearly the different rates of convergency of the literal powers in 
the three series, it will be sufficient to consider the distance of the two planets of which 
the perturbations are to be computed, as ==: (RR+rr — zR rxc, z), where R and r 
denote their mean distances from the sun, of which R is the greater, and c, z the cosine 
of the angle of commutation. Then will M.De la Grange’s series converge by the 
powers of the quantity - f ; and, since RR r r — a > an( ^ r ~b, in our notation> 
and the converging quantity in M. Euler’s series is (n n) — — 
r 4 a — b 
— x. — ; and c c, by the powers of which the new series converges, is ~ ■■ ~ -\ ~b ” 
( ■ YTJ 
3 — T- 2 - _ r + r r — . ( R “ r L > See the Memoirs of the Royal Academy of Sciences and 
RR-fzRr+rr (R + r) 1 
Belles-Lettres at Berlin, fori 781, p. 257 ; M. Euler’s Institutions Calculi Integrals, 
Vol. I. p. 186 ; and Art. 4, in what follows. 
