484] 
541 
484. 
ON THE VARIATIONS OF THE POSITION OF THE ORBIT IN 
THE PLANETARY THEORY. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxn. (1871—72), 
pp. 206—211.] 
It has always appeared to me that in the Planetary Theory, more especially when 
the method of the variation of the elements is made use of, there is a difficulty as 
to the proper mode of dealing with the inclinations and longitudes of the nodes, 
hindering the ulterior development of the theory. Considering the case of two planets 
to, in', and referring their orbits to any fixed plane and fixed origin of longitudes 
therein, let 9, 9' be the longitudes of the nodes, </>, </>' the inclinations (p = tan </> sin 9, 
q = tan <f) cos 9, &c., as usual); then the disturbing functions for m, m' respectively are 
developed, not explicitly in terms of cf>, <£', 9, 9', but in terms of <3>, the mutual 
inclination of the two orbits, and of 0, 0' the longitudes in the two orbits respectively 
of the mutual node of the two orbits; <4> and 0, 0' being functions (and complicated 
ones) of </>, (f>' } 9, 6'. Moreover, although in the general theory of the secular variations 
of the orbits of the planetary system, 9, $, &c., are, as above, referred to one fixed 
plane (the ecliptic of a certain date), yet in the theory of each particular planet it 
is the practice, and obviously the convenient one, to refer for such planet the 9, <£ 
to its own fixed plane (the orbit of the planet at a certain date), the effect of course 
being that </>, and consequently p, q, instead of being of the order of the inclinations 
to the ecliptic, are only of the order of the disturbing forces. It has occurred to me 
that the last-mentioned plan should be adhered to throughout; viz., that for each 
planet m, the position of its variable orbit should be determined by 9, the longitude 
of its node, and </>, the inclination in reference to the appropriate fixed plane (orbit 
of the planet at a certain date) and origin of longitude therein. The disturbing 
functions for the planets m and to' will of course depend not only on 9, 9', <£, <f>', 
but on the quantities <3>, 0, 0' which determine the mutual positions of the two fixed