[ 86 ] 



XV. On the Perturhatioyis of Planets moving in Eccentric 

 and Inclined Orbits. By Sir J. Lubbock, Bart., F.R.S. 



[Continued from page 6.] 



¥ N the last Number of the Philosophical Magazine I de- 

 ■*- scribed tables by which the development of the disturbing 

 function R is greatly facilitated. I shall now describe other 

 tables which have been calculated for me by Mr. Farley, 

 and which also facilitate the numerical solution. The ad- 

 vantages which the employment of tables presents wherever 

 they can be applied are well known. Not only the march of 

 the figures affords security against error, but the computer 

 acquires facility in such calculations systematically undertaken, 

 while the operations are more easy than they would be if the 

 quantities required were not connected by a common origin, 

 or so troublesome as they would be if undertaken by different 

 individuals, or by the same individual at different times. The 

 use of tables is out of the question in a literal or algebraic 

 development; but, on the contrary, it is an important pro- 

 perty of the numerical development that it can thus be mate- 

 rially facilitated. 



All developments whatever may be resolved into three 

 classes, which I call literal, quasi-literal, and arithmetic. Li- 

 teral or algebraical are those which result when the numerical 

 values of the constants are inserted last, and after the deve- 

 lopment is complete. Quasi-literal are those which result 

 when either a part only of the constants are expressed by means 

 of general symbols, or when the development is made up of 

 several distinct processes, and when the numerical values are 

 inserted after a portion of these, but not all have been accom- 

 plished. Finally, arithmetic or numerical developments are 

 those which result when the numerical valuesof the constants 

 are inserted in place of the general symbols before any step 

 of the development is attempted. 



A literal development is generally preferable, for this rea- 

 son, that if it can be performed, the development which re- 

 sults serves for every possible value which can be assigned to 

 the constants. Such, for instance, is the development of the 

 disturbing functiondue to M. Binet; and if such a development 

 in terms of the requisite variables could be accomplished and 

 carried out to a sufficient extent, and if, being accomplished, 

 numerical values of the constants could be easily introduced, 

 it would be preferable to any other. M. Hansen's develojj- 

 ment, in his Memoir on the Perturbations of Encke's Comet 

 by Saturn, is a quasi-literal development, because a portion 

 only of the processes is general. The conversion of the quan- 



