216] 
THEORY OF ELLIPTIC MOTION. 
473 
The foregoing tables are read as follows: for instance, in the Table (x°, ad, x~, xf) . ' f, 
sm 
p. 376, et seq., the third and fourth lines of the «°-compartment show that 
cos f— — e 
+ (i + F^ 4 
+ ( -§e 3 
+ &c. 
) 2 cos g 
— 4L e 7 ) 2 cos 2g 
tMìM 6 ) 2 sin g 
~ tVïï e 7 ) 2 sin 2g 
+ &c. 
and the first and second lines give the sum and difference respectively of the corre 
sponding coefficients of the cosine and sine series. 
Addition, 28th Dec. 1860.—The tables have been verified by me, on the proof-sheets; 
; 1-tables, and a portion of the sine lines of 
'/ 
the «-tables, the cosine lines of the 
the same tables, in the manner explained at the commencement of the memoir; but 
for the remainder of the sine lines it was found easier to employ the following mode 
V 
of verification ; viz. the equation - = 1 + x, gives 
{GD - 4 CJ+ 6 (ïï - 4 (5)} sin ^ =( - 1+ ^ sin ^ ; 
and as regards the terms up to e 7 , the limit of the tables, 
sin jf= (1 — « 5 + 5« 6 —15« 7 ) sin jf 
which equations afford the verification referred to. In going over the earlier sheets 
I omitted to see that the fractions were in their least terms, and it may happen 
that, in some instances, they are not so. [A few reductions have been made, and I 
believe that the fractions are now all in their least terms.] 
The expression for the true anomaly / itself has been repeatedly calculated to a 
much greater extent, in particular by Schubert {Astronomie Théorique, Pet. 1822), as 
far as e 20 . The easiest way of obtaining it seems to be by means of the equation 
C. III. 
60