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IV. On the Determination of the Terms in the Disturbing Function of the fourth order 

 as regards the Eccentricities and Inclinations which give rise to Secular Inequali- 

 ties. By J. W. Lubbock, V.P. and Treas. R.S. 



Received October 16, — Read November 20, 1834. 



Hitherto in the theory of the secular inequalities the terms in the disturbing 

 function of the fourth order as regards the inclinations have been neglected. As the 

 magnitude of these terms depends, in great measure, upon certain numerical co- 

 efficients, it is impossible to form any precise notion a priori with respect to their 

 amount, and as to the error which may arise from neglecting them. I have therefore 

 thought it desirable to ascertain their analytical expressions ; and the details of this 

 calculation form the subject of this paper. Some of the secular inequalities which 

 result from these terms are far within the limits of accuracy which Laplace appears 

 to have contemplated in the third volume of the M^canique Celeste. 



The method which I have here adopted for developing the disturbing function rests 

 upon principles which I have already explained *. Very little trouble is requisite to 

 obtain certain analytical expressions for the terms upon which the secular inequalities 

 depend, or for any others, in the development of the disturbing function ; but it is not 

 so easy to put these expressions in the simplest form of which they are susceptible ; 

 and this is a point to which I think hitherto sufficient attention has not been paid. 

 It will be found that I have obtained, finally, expressions of very remarkable sim- 

 plicity : to accomplish this, however, I have been obliged to go through tedious pro- 

 cesses of reduction, the details of which are here subjoined, in order that my results 

 may be verified or corrected without difficulty. In order to give an additional example 

 of the great facility with which terms in the disturbing function are arrived at by my 

 method, I have calculated one of those given by Professor Airy, and which is required 

 in the determination of his inequality of Venus ; and I have arrived at the result which 

 he has given. The same method, with certain modifications, is applicable to the de- 

 velopment of the disturbing function in terms of the true longitudes. The terms in 

 the disturbing function which give rise to the secular inequalities of the elliptic con- 

 stants, when the terms of the order of the fourth powers of the eccentricities and 

 inclinations are retained, and higher powers of those quantities are neglected, are as 

 follows : and I propose, as they form, in fact, a system apart, to distinguish them by 

 the indices given in the left-hand column. 



* Philosophical Transactions, 1832, Part II. 

 MDCCCXXXV. I 



