THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 41 



half of the time ; and since a double force acting during one-half the time produces 

 the same effect as a single force acting during a double interval, it follows that the 

 magnitude of the resulting inequalities will remain unchanged. 



23. The rigorous determination of the separate effects of the corrections of the 

 masses fi, fi, ft", &c. on the values of the constants which determine the secular 

 variations of the elements, when the masses simultaneously vary, is a much more 

 difficult and intricate problem than that of the determination of the secular inequali- 

 ties themselves. For, if we employ the masses in indeterminate forms, m(l-\-ft), 

 m'(l-f ft), m"(l-\-ft"), &c., instead of m, m', m", &c., it is evident that the solutions 

 of the differential equations would contain terms depending on ft, ft', ft", &c., and 

 on all the powers and products of these quantities up to [S, ft 1 , ft" 7 , inclusive, in 

 addition to the terms already calculated. And if we neglect the powers and pro- 

 ducts of ft, ft, ft", &c., above the first, it is evident that our solution would be very 

 imperfect, unless fi, ft', fi", &c. were very small quantities. Unfortunately, this is 

 not the^case, for the masses of some of the planets are still very imperfectly known; 

 and consequently the terms depending on the powers and products of ft, ft, fi", &c. 

 ought not to be neglected. There seems to be only one practicable method of 

 determining the effects of the corrections of the masses on the values of the con- 

 stants which we have already determined. And this method consists in supposing 

 the mass of each planet, in succession, to be increased by a finite quantity, //, and 

 then determine anew all the constants in the same manner as with the assumed 

 masses. If we then subtract the values of the constants which depend on the 

 assumed masses, from the values of the constants which result from the corrected 

 mass of the planet, we shall obtain the coefficient of the correction depending on 

 ft, by dividing the difference of the constants by fi , or the finite variation of the 

 mass of the planet. In this way we get the whole variation resulting from the 

 assumed variation of mass, or, in other words, we retain the terms depending on 

 all the powers ft , fi 2 , jt/ 3 , &c., and neglect only the terms depending on the products 

 ft, ft, ft", &c., when they simultaneously vary. As this is the method which we 

 have adopted, we shall here give the resulting fundamental equations, together with 

 the values of the constants determined by their solution. 



24. We shall now suppose the mass of Mercury to be increased to two and one- 

 half times its assumed value. In this case fi a will be equal to 1|, and OT 



4H6575l~ 1946300 4' value of the mass of Mercur y is ver y near ty tne 



same as that employed by astronomers, during the early part of the present century, 

 and is doubtless considerably larger than the actual value. But as the perturbations 

 produced by this planet are very small, a considerable variation of its mass will 

 produce only a small variation in the values of the fundamental equations. We 

 shall now compute the effect which this change in the mass of Mercury produces 

 in the fundamental equations. 



6 October, 1871. 



