1893.] Elasticity of a Crystal according to Boscovich. 71 



meaning of 0'(r). To render the solid, constituted of our homo- 

 geneous assemblage, elastically isotropic, we must, by 19 (24), have 

 A B = 2n, and therefore, by (32), 



Tff Q 2 Wi . , (34) . 



26. The last three of the six equilibrium equations 16 (18) are 

 fulfilled in virtue of symmetry in the case of an equilateral as- 

 semblage of single points whatever be the law of force between them, 

 and whatever be the distance between any point and its nearest 

 neighbours. .The first three of them require in the case of 23 

 that F(X) = ; and in the case of (24) that F(X^/2) = 0, results of 

 which the interpretation is obvious and important. 



27. The first three of the six equilibrium equations, 16 (18), 

 applied to the case of 25, yield the following equation : 



yiF(xy2) = -F(X) (35); 



that is to say, if there is repulsion or attraction between each point 

 and its twelve nearest neighbours, there is attraction or repulsion 

 of x/2 of its amount between each point and its six next-nearest 

 neighbours, unless there are also forces between more distant points. 

 This result is easily verified by simple synthetical and geometrical 

 considerations of the equilibrium between a point and its twelve 

 nearest and six next-nearest neighbours in an equilateral homogeneous 

 assemblage. The consideration of it is exceedingly interesting and 

 important in respect to, and in illustration of, the engineering of 

 jointed structures with redundant links or tie- struts. 



28. Leaving, now, the case of an equilateral homogeneous as- 

 semblage, let us consider what we may call a scalene assemblage , 

 that is to say, an assemblage in which there are three sets of parallel 

 rows of points, determinately fixed as follows, according to the system 

 first taught by Bravais : * 



I. Just one set of rows of points at consecutively shortest dis- 

 tances \i. 

 II. Just one set of rows of points at consecutively next-shortest 



distances \2. 



III. Just one set of rows of points at consecutive distances shorter 

 than those of all other rows not in the plane of I and II. 



To the condition AS > X 2 > \ we may add the condition that none 

 of the angles between the three sets of rows is a right angle, in 

 order that our assemblage may be what we may call wholly scalene. 



* ' Journal de 1'Ecole Poly technique,' tome xix, caliier xxxiii, pp. 1 128 : Paris, 

 1850. 





