a Problem in physical Astronomy . 
545 
of the fluents on the right-hand side, when z=7r, will become 
barely C z = C tt. Therefore, the fluent of the left-hand side 
of the equation, when z = i r, or of x ~_ ^ x) - , when 
v = i, will be = C7r. The fluent of this fluxion, it is evident, 
will consist of three parts, the first and second of which, n be- 
ing = J-, are obviously attainable from the values of A and B 
above found in Art. 9. and 16. ; and the third in series similar td 
those which have been given in the former part of this paper. 
It is evident also that, if n be = f, all three parts of this fluent 
are attainable from the values of the two coefficients already 
found, and C' would be = — 2 A' -f ~ ( B'a — B ) . 
lg. And in this manner may the other coefficients, D, E, 
F, &c. be determined. And since the cosines of 3 z, 4 z, &c. 
are = 4 a: 3 — gx, 8a: 4 — 8o: 2 -f- 3, &c. respectively; and since 
x = ., it is evident that the numerator of the fraction 
into which the fluxion in the preceding article is to be multi- 
plied, will be always of this form, viz. p-\-qvv-\-rv'-\- sv *> &c-\ 
from which it follows, that, if the values of A', A, A, &c. cor- 
responding to n, n — 1, n — 2, &c. be computed, the values of 
C, D, E, F, and all the rest, may be found in terms of A', A, 
A, &c. with the coefficients a and b. But, since the easiest 
method, that has come to my hands, of computing the values 
of C, D, E, &c. after A and B are found, is explained in M. 
Euler’s Institutiones Calculi integralis, Vol. I. p. 181,* I shall 
* The coefficients 2, 3, and 4, after zC sin. 3 D sin. and 4E sin. in line 8 of the 
page above referred to, are wanting; and — is printed for -f- before 2 C, in line 13, 
And there are press errors in many other places. It is to be regretted, that so excel-,, 
lent a book was not more correctly printed. 
MDCCXCVIIL 4 A 
Or of 4 aa —2bb~$a [a + b)vv + ^{ a + b^v 4 - 
bb{a-\-b) n 
, when 
