TRANSACTIONS OF SECTION A. Gl9 



be ascertainable from the morphological constants unless the degrees of resistance 

 presented in difterent directions are known ; except, however, the cases of the 

 more symmetrical systems in which the positions of these axes are fixed by 

 symmetrical considerations. 



_A similar observation applies to the absorption-figure for monochromatic light, 

 which is also an ellipsoid. 



The fact that the elastidty-Jigure of crystals is a surface of a higher order than 

 an ellipsoid is due to its being the outcome of a compounding and averaging whose 

 scope is more limited and not so uniform as that above referred to. 



11. On a Species of Tetrahedron the Volume of any memher of which can 

 be determined without employing the proof of the proposition that 

 Tetrahedra on equal bases and haviiig equal altitudes are equal, 

 which dejjends on the Method of Limits. By M. J. M. Hill, M.A., 

 D.Sc.,F.B.S., Professor of Mathematics at University College, London. 



The object of this communication is to prove the existence of the species of 

 the tetrahedron mentioned in the title. 



Art. 1. A proof is first given of the known proposition, that if the edges B A, 

 C A, D A of the tetrahedron A B C D be produced through A to E, F, G respec- 

 tively, so that BA = AE, CA = AF, DA = AG, then the tetrahedra ABCD, 

 A E F G are of equal volume. 



Art. 2. From the above proposition it is deduced that if the edge D A of the 

 tetrahedron A B C D be perpendicular to the plane ABC, and if D A be produced 

 to E, so that D A = A E, then the tetrahedra ABCD, A B C E are of equal 

 volume. 



Art. .3. Now let A B C D be a tetrahedron, and let D H, C K be drawn equal 

 and parallel to B A. 



JoinHA, AK, KH, HC. 



Then if B H be perpendicular to the plane A C D, it follows, bv applying Art. 2 

 twice over, that the tetrahedra A B C D, A U C H are of equal vo'lume. 



In lilv-e manner if D K be perpendicular to the plane A H, it follows that the 

 tetrahedra A D C H, A H C K are of equal volume. 



Hence the tetrahedron ABC D is one third of the prism, havino- the same base 

 and altitude. " 



The two conditions — 



(1) That B H is perpendicular to the plane A C D, and (2) that D K is perpen- 



