2] ON THE PERTURBATIONS OF URANUS. 39 



For the observations of the last two years, I am indebted to the 

 kindness of the Astronomer Royal. The three years nearly agree in shewing 

 that the errors of the first hypothesis are to those of the second in the 

 ratio of 5 to 4, from which I inferred, in a letter to the Astronomer Royal, 



dated September 2, 1846, that the assumption of 7 =sin 35 = 0'574, would 

 probably satisfy all the observations very nearly. 



54. The results which I have deduced from Professor Challis's obser- 

 vations of the planet, strongly confirm the inference that the mean distance 

 should be considerably diminished. It is of course impossible to determine 

 precisely, without actual calculation, the alteration in longitude which would 

 be produced by such a diminution in the distance. By comparing the values 

 of 6 given by the two hypotheses, it may be seen, however, that if we took 

 successively smaller and smaller values for the mean distance, the values 

 found for the mean longitude in 1810 would probably go on diminishing, 

 while at the same time the mean motion from 1810 to 1846 would rapidly 

 increase, so that the corresponding values of the mean longitude at the 

 present time would probably soon arrive at a minimum, and afterwards begin 

 again to increase. This I believe to be the reason why the longitude found 

 on the supposition of too large a value for the mean distance agrees so 

 nearly with observation. In consequence of not making sufficient allowance 

 for the increase in the mean motion, I hastily inferred, in my letter to the 

 Astronomer Royal mentioned above, that the effect of a diminution in the 

 mean distance would be to diminish the mean longitude. 



55. I have already mentioned, that I thought it unsafe to employ 

 Flamsteed's observation of 1690 in forming the equations of condition, as 

 the interval between it and all the others is so large. The difference between 

 it and the theory appears to be very considerable, and greater for the second 

 hypothesis than for the first, the errors being + 44 //- 5 and + 50"'0 respec- 

 tively. These errors would probably be increased by diminishing the mean 

 distance. It would be desirable that Flamsteed's manuscripts should be 

 examined with reference to this point. 



56. The corrections of the tabular radius vector of Uranus may be 

 easily deduced from those of the mean longitude by means of the following 

 formula : 



