518 ON SOME FORMULAE RELATING TO THE VARIATION OF A PLANET’S ORBIT. [219 
then co, T, <E>, are given in terms of cr — <r 0 , cf) 0 , cj); and we have, moreover, XG = 0 + AC 
= 0 o + co + T, X 0 G = (x 0 + co; that is, the position of the travelling orbit X 0 X, and origin 
of longitudes X 0 therein, are determined by 
0 O + (o + r, the longitude of node, 
cr 0 + « , the departure of node, 
, the inclination. 
Suppose that in reference to this travelling orbit and origin of longitudes therein, 
we have 
v', the longitude of planet, 
y', the latitude of ditto, 
viz., in the figure X 0 N=v' (and therefore BN = v' — 0 o ), PX—y'. 
Moreover, BG + CM = BG + AM — AG = co +(v — 0) — (0» — 0 + (o + Y) = v — 0 O —Y, hence 
the two equations are 
cos y sin (v — 0 n — T) = cos y' sin (v' — 0 O ) — tan \ <3> cos w (sin y + sin y'), 
COS y COS (v — 0 O — r) = COS y' COS (V' — 0 O ) + f an 2 ^ S ^ n w ( S ^ n y + s i n y')> 
or, as they may also be written, 
cos y sin (v — 6 0 — T) = cos (f>o sin (J? — cr 0 ) — tan £ cos co (sin y + sin y'), 
cos y cos (v — 0 O — T) = cos (]? — cr 0 ) + tan ^ <I> sin <w (sin y + sin y'), 
or, if we put s = sin y + sin y, then observing that sin y =■= sin cf> sin ty — a), sin y' = 
sin </) 0 sin (J> — <r 0 ), these become 
cos y sin (v — 0 O — T) = cos (f> 0 sin (J? — a- 0 ) — tan \ <J> . s cos co, 
cos y cos (v — 0 O — T) = cos (}? — <r 0 ) + tan ^ . s sin w, 
sin y {= sin </> sin (J? — cr)} = — sin (f>o sin (]? — cr 0 ) + s, 
which are, in fact, Hansen’s formulae (16), p. 75, the letters corresponding as follows, 
viz., 
v, ]?, y, a, cr 0 , 0, 00, (f), 00 , <j>, r, (O (supra) to 
l, v, b, cr, h , 0, h , i , —k , 2y, T, co (Hansen). 
where, of course, the correspondence </> 0 to — k, shows that these angles are measured 
in a contrary direction. I had from Hansen’s equations expected that the above formulae 
would have contained sin y — sin y in place of sin y + sin y'. 
2, Stone Buildings, W. C., 4th Dec, 1860.