694 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



we multiply the difference (1,58:^) between any two consecutive 

 planets' velocities, by the greater serial number of the two, the 

 product will be the greater velocity of the two. ^he reason is 

 that the velocities and their differences are in a definite ratio to 

 each other. 1,582 is one-half the velocity of Nox, and the same 

 number (1,582) is one-third the velocity of Cer])erus, and one- 

 fourth that of Bacchus, and so on through the whole series. Now, 

 when we examine the corresponding series of square roots ot the 

 distances in table 9, we and that the same ratio exists there: that 

 is to say, if we multiply the difference between any two consecu- 

 tive square roots by the greater serial number of the two, the pro- 

 duct will be the greater square root of the" distance of the two. 

 The reason is as follows: the velocities and their differences, 

 havinsr a certain regular ratio to each other, the square roots of 

 the distances and their differences (being in proportion to the 

 velocities) have the same ratio to each other. Once more I remark 

 that I wish the critical reader to bear in mind, that the design of 

 these tables is, not to show how near the theoretical velocities or 

 square roots agree with the actual, but to exhibit the remarkable 

 relations Avhich exist l)etween the velocities and the square roots, 

 arranged in what I conceive to be tJieir true serial order. I also 

 wish to furnish the most decisive proofs that the relations which 

 I have pointed out are not mere coincidences, but are founded 

 upon a law heretofore unknown. 



TABLE 8. 



a ' ». I ni u 



O . a . S S ® 3 



ill S g|i S-5^ 



.'i o o -1 ^ tS .- Or a°Q< 



oC^ ^ -^ „ OZZ fl— •" 



Chaos 1,582 x 1 = 1,582 



Difference 1,582 " 2 " 3,164 



Nox. 3,164 



Difference 1,582 " 3 " 4,746 



Cerberus 4,746 



Difference 1,582 " 4 " 6,328 



Bacchus 6,328 



Difference 1,582 " 5 " 7,910 



Janus 7,910 



Difference 1,582 " 6 " 9,492 



Atlas 9,492 



