218 Proceedings of Royal Society of Edinburgh. [sess.. 
be placed for equilibrium ; and after that try to find whether the 
equilibrium is stable or unstable. 
§ 24. Calling /(D) (as in § 4) the attraction between atom and 
atom at distance D, we have for the sum, P, of attractions between 
all the atoms on one side of any point in their line, and all the- 
atoms on the other side, the following finite expression having 
essentially a finite number of terms, greater the smaller is a : 
/(a) + 2/(2a) + 3/(3a)+ . . . . =P . . . (8). 
Hence a, for equilibrium with no extraneous force, is given by the 
functional equation 
/(a) + 2/(2o) + 3/(3a)+ .... =0 . . . (9) 
which, according to the law of force, may give one or two or any 
number of values for a : or may even give no value (all roots 
imaginary) if the force at greatest distance for which there is force 
at all, is repulsive. The solution or all the solutions of this 
equation are readily found by calculating from the Boscovich 
curve representative of /(D) a table of values of P, and plotting 
them on a curve, by formula (8), for values of a from a = I (the 
limit above which the force is zero for all distances) downwards 
to the value which makes P = — go , or to zero if there is no 
infinite repulsion. The accompanying diagram, fig. 6, copied 
from fig. 1 of Boscovich’s great book,* with slight modifications 
(including positive instead of negative ordinates to indicate 
attraction) to suit our present purpose, shows for this particular 
curve three of the solutions of equation (8). (There are obviously 
several other solutions.) In two of the solutions, respectively, 
A 0 , A', and A 0 , A", are consecutive atoms at distances at which the 
force between them is zero. These are configurations of equi- 
librium, because A 0 B, the extreme distance at which there is 
mutual action, is less than twice A 0 A', and less than twice A 0 A". 
In the other of the solutions shown, A 0 , A p A 2 , A 3 , A 4 , A 5 , A 6 
are seven equidistant consecutive atoms of an infinite row in 
* Theoria Philosophise Naturalis redacta ad unicam legem virium in 
natura existentium, auctore P. Rogerio Josepho Boscovich, Societatis Jesu,. 
nunc ab ipso perpolita, et aucta, ac a plurimis prsecedentium editionum 
mendis expurgata. Editio Yeneta prima ipso auctore prsesente, et corrigente. 
Yenetiis, MDCCLXIII. Ex Typographia Remondiniana superiorum per- 
missu, ac privilegio. 
