1896.] certain Properties of the Tetrahedron. 101 



The characteristic equations of mm and mm may readily be 

 deduced from the catena of relations satisfied by m and m'. They 

 will be found to be 



(mm') 2 - 2mm' {2 Urn Um' - Lmm'} + ( Wm . Wm') 2 = 0, 

 (mm)- — 2m'm [2 Um Um — Lmm'} + ( Wm . Wm') 2 = 0. 



Hence we have 



Umm = Um'm = 2 Um Um' — Lmm' (15), 



and Whim' = Wm'm = Wm . Wm' (16). 



We must here take notice of a peculiarity of the symbol L, 

 which does not belong to the other operative symbols we have 

 made use of, viz. that L (mm' . m") is not equivalent to 



L (m . mm"), 



nor may we replace L (mm' . co) by Lmm'. This is evident from 

 the definition of L, and may also be brought out with the aid 

 of equation (15), which may be made to furnish us with ex- 

 pressions for the two quantities to be compared. 



Finally, we have 



Km' . Km = (2 Um' - m') (2 Um - m) 



= 4 Um Um' — 2m Um — 2m Um + mm 

 — 4 Um Um — 2Lmm — mm' 

 = 2 Umm' — mm' 

 = Kmm'. 



3. The matrices that we shall make use of in this paper will, 

 for the most part, be expressed in a special form. We will suppose 

 p and q to be two constant matrices satisfying the catena of 

 relations 



ap 2 -2hp + b = 0, 



aq 2 — 2gq + c = 0, 



a (pq + qp) — 2gp — 2hq + 2/= 0, 



where a, b, c, /, g, h are scalar constants ; and the matrices with 

 which we shall be chiefly concerned will be of the form 



m = x + py + qz, 

 where x, y, z are scalars. 

 Now we have 



aUp = h, aUq = g, a( Wp) 2 = 6, a ( Wq) 2 = c, 

 aLoop = aLpco = aUp = h, 

 aLcoq = aLqco =aUq = g, 

 aLpq = aLqp =f. 



