247 



trived to eliminate them, but by a process so long and com- 

 plicated, that it is extremely difficult to follow him through 

 the numerous transformations he has employed, and to see 

 the reasons of them. This part of the theory is greatly sim- 

 plified in the memoir, by developing only relatively to the 

 quantities which it is our object to get rid of: we, can deve- 

 lope with reference to the other quantities equally well, and 

 even with advantage afterwards. The developments in ques- 

 tion are effected with regard to the powers of the disturbing 

 force, and not in series of sines and cosines. 



It is easier to find the radius vector and longitude on the 

 plane of the orbit than by referring the motion to a fixed 

 plane ; but it is a matter of more difficulty to find the latitude 

 in this case. In this part of the theory the author has intro- 

 duced changes which may be employed very advantageously 

 when the inclination is small. The inclination (i) and the 

 longitudes (^ and 9) of the node on the two planes produce 

 many terms in the disturbance function. In place of these 

 quantities, the latitude and reduction (the two sought quanti- 

 ties) are, in the memoir, introduced into this function, and 

 into its partial diflPerential coefficients. Afterwards sin i and 

 sin (y - 6) are obtained, instead of the three quantities, t, 9, ^, 

 or p, g, and ^-9, which M. Hansen finds. Thus, the lati- 

 tude is found as readily as when the motion is referred, in the 

 first instance, to a fixed plane. 



The memoir concludes with a transformation of the diffe- 

 rential equations from a fixed to a sliding plane, preserving a 

 constant inclination (that of the orbit) to the fixed plane, and 

 sliding upon it with a uniform motion, equal to the mean mo- 

 tion of the node. No details are added here ; the transforma- 

 tion is merely noticed as appearing to merit further conside- 

 ration. 



Rev. W. Roberts read a paper, from which the following 

 is an extract. 



