72 Lord Kelvin. On the [June 15,. 



29. Let A'OA, B'OB, C'OC be the primary rows thus deter- 

 minately found having any chosen point, O, in common ; we have 



A'O - OA = 



B'O = OB = X 2 )> (36). 



C'O = 00 = X 3 _ 



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

 next-nearest neighbours ; and C' and C, 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 O than B 

 and B', are nearer to O 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 di- 

 chirally similar* tetrahedrons of each 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 distances of O from any other of the points on 

 the two sides of the plane AOB.f 



31. As a preliminary to the engineering of an incompressible 

 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 tetrahedron whidh fulfils this condition 

 has found a place in geometrical treatises, but I am informed by Dr. 

 Forsyth that it has appeared in Cambridge examination papers. For 

 my present purpose ifc occurred to me thus : Let QO, 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 tetra- 

 hedron OABC is a maximum. Whatever be the four given lengths r 

 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 tetra- 

 hedron in general are required to make it our special tetrahedron. 



32. Hence we see the following simple way of drawing a special 

 tetrahedron. Choose as data three sides of one face and the length 



* Either of these may be 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), (A), (i), 'Math, and Phys. 

 Papers,' vol. iii, pp. 412 413. 



