THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 61 
The expressions for cos ¢, sine, are the same as those of cos ¢’, sine’, if we omit 
the accents. 
Hence if we perform the operations indicated in the expression for (47), we have 
ay (2) 
Tr a 
= 30” [hy yy + Wd; Wy] cos (41g — Vg!) — 30? [By Ely 0'v] sin(Lig—’g’) (2) 
é and v’ having all positive values. 
Attributing to 7 and 7 particular values, we find, noting that 6, = 0, and 0’, = 0’, 
(1) = 3 [heyy + W004 Jeos( g— 9')— 3 [byi t+ lydi] sin( g— 9’) 
+ $3 [h.nyi—h'd,0, ]cos(—g— 9’) — [By — 7184] sin (—g— g’) 
+ gh. cos ( — J)— flys, sin( — g’) 
+ 2 [h. yy’, +h’. 8,82] cos( g—2q’) — 2 [h.by’.4+ U'y,8'2] sin( = g — 2g’) 
+ 2 [h.yiy’2—h’. 30’. cos (— g — 29’) — 2 [1.d,7’.— V'y,0’.] sin (— g — 29’) 
+ 2h.yuy's Os ( — 2q') — 20. yoo sin ( — 29’) 
+ 3 [h.yy’3+ h’.60'3] cos(  g—3q’) — $[0.by7/3+U.70'3] sin( g—38g’) 
+ ete. — ete. 
+ $ [h.yoy'1 + W’.6,0.] cos( 2g— 9’) — 4[bb.y. 40.7281] sin ( 29 — 7’) 
+ 3 [he yoy + h'.630',] cos (— 29 — 9’ 3 [Ld —U.y20',] sin (— 2g — 9’) 
+ ete. — ete. 
The numerical value of (#7) given by (1) must first be transformed into a series 
in which both the angles involved are mean anomalies before it can be compared with 
the value given by the equation just found. 
If we find the value of (H) from the preceding equation, it can be checked by 
means of the tables in BrssEv’s Werke. 
The expression for « (“) is known; and with the expression for (/7) just given, 
we obtain the value of 
Ce On" i (“) —(). 
The next step is to obtain expressions for the disturbing forces. 
