OF ARTS AND SCIENCES. 67 



Sun in 210 years, which is exactly two and a half times the period of 

 the revolution of Uranus. Now, if the times of revolution of two 

 planets were exactly as 2 to 5, the effects of their mutual influence 

 would be peculiar and complicated, and even a near approach to this 

 ratio gives rise to those remarkable irregularities of motion which are 

 exhibited in Jupiter and Saturn, and which greatly perplexed geome- 

 ters until they were traced to their origin by Laplace. This distance 

 of 35.3, then, is a complete barrier to any logical deduction, and the 

 investigations with regard to the outer space cannot be extended to the 

 interior. 



" The observed distance 30, which is probably not very far from the 

 mean distance, belongs to a region which is even more interesting in 

 reference to Uranus than that of 35.3. The time of revolution which 

 corresponds to the mean distance 30.4 is 168 years, being exactly 

 double the year of Uranus, and the influence of a mass revolving in 

 this time would give rise to very singular and marked irregularities in 

 the motions of this planet. The effect of a near approach to this ratio 

 in the mean motions is partially developed by Laplace, in his theory 

 of the motions of the three inner satellites of Jupiter. The whole per- 

 turbation arising from this source may be divided into two portions or 

 inequalities, one of which, having the same period with the time of 

 revolution of the inner planet, is masked to a great extent behind the 

 ordinary elliptic motions, while the other has a very long period, and 

 is exhibited for a great length of time under the form of a uniform in- 

 crease or diminution of the mean motion of the disturbed planet. But 

 it is highly probable that the case of Neptune and Uranus is not mere- 

 ly that of a near approach to the ratio of 2 to 1 in their times of rev- 

 olution, but that this ratio is exactly preserved by those planets ; for it 

 may be shown, as was shown by Laplace for the ratio two fifths, that 

 a sufficiently near approach to it must, on account of the mutual action 

 of the planets, result in the permanent establishment of this remarkable 

 ratio. Thus, if 



V = the mean longitude of Uranus, 



v' = that of Neptune, 

 V z= 2u' — V ; and if D expresses the differential coefficient relatively 

 to the time, a near approach to the ratio of 2 to 1 gives the equations, 

 Wv^psm. {2v'—v-^A)=psm. (V-f-A), 

 1)%' = q sin. {2v'—v-\-A) = q sin. (V -f A) ; 



