of the Satellites from their Primaries. 179 



Suppose a = 1312, a + i = l/'20, r=1.53. 



Ejnpirical Values. True Values. Differences. 



Then a = 1312 1312 o 



a+b =1720 1720 o 



a + rb = 193G 1984 + 48 



a+r"b=2'267 2275 + 8 



a+r'b= 2773 not observed 



a + r'^b = 3547 not observed 



a +r=6= 4731 4551 -180 



a + r^b = 6543 not observed 



a +r''b= 9315 9101 -214 



It is unnecessary to go through the same process for the planets. 

 I will only observe that the ratios of consecutive di.stances 

 diminished by the least distance, are in order, 1.823, 1.854, 2.066, 

 2.051, 1.900, 2.054, and the mean value = 1.958. 



5. The foregoing investigation has answered the purpose in- 

 tended, which was to shew the general coincidence of the arrange- 

 ment of the satellites with an empirical law. When a law, 

 recommending itself by its simplicity, has been establi.shed by 

 observation, and is found to be attended with considerable devia- 

 tions, we are naturally led to enquire whether the deviations 

 themselves are .subject to any law. The following process may 

 in some degree answer this end in regard to the law before us. 

 As any three distances would suffice to determine values of a, b, 

 and r, and different values of these quantities would be obtained 

 from every three that may be fixed upon, I have combined the 

 equations in all the ways they admit of, and taken the mean of 

 the different values. Substituting the mean values in the pro- 

 gression a, a + b, a + rb, &c. and comparing the distances thus 

 obtained with the true distances, if any of the differences com- 

 pared with the distances should be found much greater or much 

 less than the others, we shall be directed to some peculiarities 



