368 PROCEEDINGS OF THE AMERICAN ACADEMY § 4 



or liquid and in the gaseous state; but we know that it is 

 impossible for a given atom to be surrounded by more than 

 twelve at equal distances from it and from each other, and 

 that, in order to have thirteen or more, some of the central 

 distances must be increased. In any compact atomic ar- 

 rangement, we shall therefore not commit a serious error 

 by assuming that there are twelve atoms at unit-distance 

 from a given atom, forty-eight at two units' distance, and in 

 general 12 n^ at n units' distance. 



Not knowing how these atoms may be arranged, we 

 must apply the theory of probabilities to determine the 

 potential at any centre due to all the surrounding atoms; 

 and we shall find that this potential, like the probable 

 error of the mean of a number of terms, is proportional 

 to the square root of that number; 11^ atoms at ;/ units' 

 distance will therefore through interference have a prob- 

 able effect only as great as n atoms combined; and since 

 the potential must vary, cccteris -paribus, inversely as the 

 cube of the distance, the effect of 12 it" atoms at n units' 

 distance, as in the case of an indefinite number of magne- 

 tized particles, w411 be equal to that at unit-distance of 12 

 atoms divided by if. The whole number of atoms sur- 

 rounding a given atom will therefore be equivalent, very 

 nearlv, to 



12(1 + JL + JL + J^ _[- etc.) 



atoms at unit-distance. 



This series may be broken up into two, namely, 



' + 7+7 + 7 + ^'- 



which is shown* to be equal to —-, and a second series, 



8 



Ricmann's "Partielle Differentialgleichungen," } 23, fin. 



