40 SECULAR VARIATIONS OF THE ELEMENTS OP 



The mean annual motions of the perihelia of the four planets Jupiter, Saturn, 

 Uranus, and Neptune being 



Jupiter and Uranus 3".7166075; 



Saturn 22 .460S479 ; 



Neptune 0.6166849; 



it follows that the mean motion of Saturn's perihelion is very nearly six times that 

 of Jupiter and Uranus ; and this latter quantity is very nearly six times that of 

 Neptune ; or, more exactly, 985 times the mean motion of Jupiter's perihelion is 

 equal to 163 times that of Saturn; and 440 times the mean motion o.f Neptune's 

 perihelion is equal to 73 times that of Jupiter and Uranus. It also follows that 

 the perihelion of Saturn performs a complete revolution in the heavens in 57700 

 years ; that of Jupiter and Uranus in 348700 years ; while that of Neptune requires 

 no less than 2101560 years to perform the circuit of the heavens. 



22. Having determined the numerical values of all the constants which enter into 

 the integrals of the differential equations (A), corresponding to the assumed masses, 

 it now remains to determine the changes which would be introduced into these con- 

 stants, on the supposition that the masses were changed from m to m(\-\-y), in' to 

 m '(l+^'), &c, or, on the supposition that the masses of all the planets were multi- 

 plied by a factor (1+a), a being any supposed correction of the mass of the planet, 

 and greater than — 1. If we suppose a, or u, <u', (j.", &c to be equal to — 1, it is 

 evident the whole system of differential equations would vanish. The effect of 

 changing all the masses, in the ratio jof 1 to 1+a, on the equation of the eighth 

 degree, would be equivalent to multiplying the coefficients of the different powers 

 of g by (1+a) 8- ", n denoting the exponent of g in the given term. Consequently, 

 the coefficient of g* would remain unaltered ; that of g 1 would be multiplied by 

 1+a; that of g 6 would be multiplied by (1+a) 2 , &c; while the term of the 

 equation which is independent of g would be multiplied by (1+a) 8 . It is evident 

 that these changes are such as would be produced by multiplying each of the roots 

 of the equation by 1+a; consequently, we shall have the following theorem: — 



If the masses of all the planets be simultaneously increased in the ratio of X td 1+a, 

 all the roots of the equation in g ivill be increased in the same ratio. 



It is also evident that, if the masses of all the planets be multiplied by 1+a, the 

 values of A, A', A", Sec, D, D, D", &c. will all be multiplied by (1+a) 2 ; and, as 

 they are all multiplied by the same quantity, it is manifest that the ratios of the 

 quantities N, N, N", Sec, will remain unaltered. And since the ratios of N, N, N", 

 Sec, remain unaltered, it is evident that tan (3 will be unchanged, and consequently 

 the values of N, N, N", Sec will not be changed. It therefore follows that, if the 

 masses of all the planets be simultaneously increased in any given ratio, the magni- 

 tudes of the secular inequalities will remain unchanged. 



To illustrate, we shall observe that if the masses of all the planets be supposed 

 to be doubled, the intensity of the disturbing forces would be doubled ; but, accord- 

 ing to the preceding theorem, the roots g, g u g. 2 , Sec would be doubled ; and conse- 

 quently the disturbing forces would operate in the same direction during only one- 



