NEWTON S PRINCIPJA. 123 



quences of the equations to the eccentricity of the pla 

 netary orbits, obtained by the investigation of the total 

 effect of the mutual actions of the heavenly bodies. There 

 results from that analysis this remarkable theorem. That 

 if the eccentricities of the different planets be called e, e , 

 ef , &c., their masses m, m , m&quot;, &c., and their transverse 

 axes a, of, a&quot;, &c., and if the integration be made of the dif 

 ferential expression for the relation between the differentials 

 of the eccentricities multiplied by the sines of the longitude 

 and the differentials of the time, and for the relation be 

 tween the differentials of the eccentricities multiplied by 

 the cosines of the longitudes and the differentials of the 

 fdesin. r , d e&quot; sin. ^ decos.^ , de^cos. 



tune &amp;gt; ( -jrr^ and -dir* ~dt~ and si 



&c., we obtain the equation e 1 . m . ^ a + e v2 , m\ ^ a&quot; + ef 2 , 

 m&quot;. Va&quot;, &c. = C* ; C being a constant quantity. Now, 

 as all the motions are in the same direction, */ a y VJ, &c., 

 are all positive. Hence, it follows that each of the quanti 



ties e . m . Va, e . m v . ^\ &c., is less than C ; and sup 

 pose at any one period the whole eccentricities e, e , e&quot;, &c., 

 to be very small, which is known to be true, C, which at 

 that period was the sum of their squares, must be very 

 small; the other quantities m, m f , &c., being wholly con 



stant, and Va, V a , &c., being invariable in considerable 

 periods of time. Therefore, it is clear that the varia 

 tion in any one of those eccentricities, as e, never can 

 exceed a very small quantity, namely, a quantity propor 



tional to VC e 2 e&quot; 2 , &c. The whole possible amount 

 of the eccentricity is confined within very narrow limits. 

 It never can for any body, whose eccentricity is e, exceed 

 a quantity equal to 



* Mec. Gel. liv. ii. ch. 6, 7. (sects 53. 55. 57, 58). 



