of a Crystal according to Boscovich. 427 



A'0 = OA=X 1 "J 

 B'0 = OB=\ 2 I (36). 



C'0 = OC=\ 3 ' 



Thus A' and A are O's nearest neighbours ; and B' and B, 

 O's next-nearest neighbours ; and C and 0, O's nearest 

 neighbours not in the plane AOB. (It should be understood 

 that there may be in the plane AOB points which, though at 

 greater distances from than B and B' ; are nearer to than 

 are C and C.) 



§ 30. Supposing, now, BOC, B'OC, &c., to be the acute 

 angles between the three lines meeting in ; we have two 

 equal and dichirally similar * tetrahedrons of which each of 

 the four faces is a scalene acute-angled triangle. That 

 every angle in and between the faces is acute we readily see, 

 by remembering that OC and OC are shorter than the dis- 

 tances of from any other of the points on the two sides 

 of the plane AOB f. 



§ 31. As a preliminary to the engineering of an incom- 

 pressible elastic solid according to Boscovich, it is convenient 

 now to consider a special case of scalene tetrahedron, in which 

 perpendiculars from the four corners to the four opposite faces 

 intersect in one point. I do not know if the species of tetra- 

 hedron which fulfils this condition has found a place in geo- 

 metrical treatises, but I am informed by Dr. Forsyth that it 

 has appeared in Cambridge examination papers. For my 

 present purpose it occurred to me thus : — Let Q0 3 QA, QB, 

 QC be four lines of given lengths drawn from one point, Q. 

 It is required to draw them in such relative directions that 

 the volume of the tetrahedron OABC is a maximum. What- 

 ever be the four given lengths, this problem clearly has one 

 real solution and one only : and it is such that the four planes 

 BOC, COA, AOB, ABC are cut perpendicularly by the lines 

 AQ, BQ, CQ, OQ, respectively, each produced through Q. 

 Thus we see that the special tetrahedron is defined by four 

 lengths, and conclude that two equations among the six edges 

 of the tetrahedron in general are required to make it our 

 special tetrahedron. 



* Either of these may he turned round so as to coincide with the image 

 of the other in any plane mirror. Either may be called a pervert of the 

 other; as, according to the usage of some writers, an object is called a 

 pervert of another if one of them can be brought to coincide with the 

 image of the other in a plane mirror (as, for example, a right hand and a 

 left hand). 



t See "Molecular Constitution of Matter," § (45), (7i), (i), 'Math, and 

 Phys. Papers/ vol. iii. pp. 412-413. 



