692 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



aminmg the square roots of the distances of these two planets, 

 we find the same proportion (though inverse) exists between them; 

 that of Mars being 12 and that of Mercury 6, which is half as 

 Biuch ; in other words, as 110 is to 55 so is 12 to 6. Again, the 

 velocity of Uranus is 15,800 miles per hour, that of Mercury is 

 110,725, which is 7 times more ; so also the square root of the 

 distance of Mercury is 6 and that of Uranus 42, which is 7 times 

 more. In the same way the mean velocities of any two planets in 

 the series may be compared with the square roots of the distances 

 of the same two planets, and found to be proportional. 



This law of inverse proportion is of great practical value, in 

 connection with our new theory ; for, when the law of common 

 difference of velocities indicates the existence of an undiscovered 

 planet, the law of proportion enables us to determine its distance 

 from the sun. For instance, in the space between Neptune and 

 Uranus, the law of common dilFereuce of velocities indicates a 

 planet, which, in the table, I have named Pluto, the velocity of 

 which is 14,238 miles per hour. Now, as the calculated velocity 

 of Pluto is to the known velocity of Mercury," so is the square 

 root of the known distance of Mercury to the answer required ; 

 that is, to the unknown square root of the distance of Pluto. 



One important result of this perfect proportion of the square 

 roots to the orbital velocities, is that some of the same serial rela- 

 tions exist between the square roots that are found between the 

 velocities. This is illustrated by the following table : 



^ Explanation of Table 7. 



The first, or left hand column of figures in table 7 is a repitition 

 of 425, which is the square root of the mean distance of Chaos, 

 the most distant possible planet in the series. This 435 is ob- 

 tained by multiplying 6,071, the square root of 36,857,000 (the 

 mean distance of Mercury,) by its serial number, 70. By succes- 

 sively dividing 425 by the serial numbers in the second column, 

 we obtain the square roots of the distances of all the planets in the 

 series, as they are represented in the third column. The fourth 

 column contains the mean distances the planets from the sun, 

 obtained by squaring the theoretical square roots in the third 

 column. The fifth column contains the actual mean distances, so 

 placed as to admit of easy comparison : 



