of the Satellites from their Primaries. 179 



cause, when once the satellites have fallen into this order, and 

 prevent an exact conformity to it. With res])ect to the other 

 satellites, the difference compared with the distance is greatest 

 for the third, and next greatest for the sixth. The sixth satellite 

 was first discovered and then the third. It is jirobable that these 

 are the two largest. 



8. Lastly, suppose for the satellites of Uranus r = %, a= 1312, 



+ 6 = 1720, rt + ri=1984, fl+r-6=2275, rt + r^t = 455l, a + r'6 = 9l01. 



From these six equations values of a and b may be found in 

 fifteen different ways. The mean value of a = 1283, that of 6 = 437. 



Empirical Values. True Values. Differences. 



Hence a = 1283 1312 + 29 



a + b = 1720 1720 O 



a + rb =1934 1984 + 50 



a + rb - 2259 2275 + 16 



a + r^6 = 4601.. 4551 — 50 



a + r'b = 8749 9101 +352 



The differences are least for the second and fourth satellites, 

 pointing them out as maximum causes of derangement. The 

 second and fourth are the satellites Herschel first discovered, and 

 they are the only ones which have since been seen' by other 

 Astronomers. The natural conclusion from this, that they are 

 the largest, renders still more probable the inference drawn from 

 the consideration of Jupiter's satellites, that the large bodies of 

 a system derange the law of distances by their gravitation. Astro- 

 nomers have doubted of the existence of the other four satellites. 

 This singular law, which the great discoverer of Uranus and of 

 his .satellites has aided in establishing, reciprocally confirms the 

 value and correctness of his observations. 



9. When we come to treat the distances of the planets in 

 a similar manner, we are presented with some difficulty by the 



Z 2 



