100 Mr Brill, On the Generalization of [Feb. 24, 



The catena of relations satisfied by m and m' may be written 

 in the form 



m 2 — 2m Um + ( Wm) 2 = 0, 



m' 2 - 2m' Um' + ( Fm') 2 = 0, 



mm' + m'm — 2m Um — 2m' ?7m + 2Lmm' = 0. 



From these we may readily deduce the characteristic equation 

 of a matrix of the form xm + ym, where x and y are scalars. 

 We have 



(xm + ym') 2 = xFm? + xy (mm + mm) + y 2 m' 2 

 = 2 (xm + ym') (x Um + y Um') 



— {x n - ( W mf + 2xyLmm' + y 1 ( Wm') 2 }. 

 Thus we obtain 



U(xm + ym') = xUm + yUml (11), 



and, as a special case, 



Uxm = xUm. 



Thus the operator U is commutative with scalars, and is also 

 distributive in its operation. It follows from equations (2) and 

 (4) that the operators K and V possess the same properties. 

 Further, the formula (11) is capable of generalization by suc- 

 cessive application, so that we obtain 



U%xm — %x Um, KXxm = XxKm, 



Vtxm = ^xVm (12). 



We also have 



{ W (xm + ym')} 2 = x 2 ( Wm) 2 + 2xyLmm + y 2 ( Wm') 2 . 



This equation also may be generalized by means of a property 

 of the symbol L, which we proceed to develope. We have 



2L (%xm . m") = (Zxm) Km" + m"K^xm 



= %x (mKm" + m"Km) 



= 22xLmm" (13). 



Also, if in (8) we write ml = m, it reduces to (1), so that we 

 have 



Lm 2 = ( Wm) 2 . 



Thus we obtain 



(WXxm) 2 = L (Stem) 2 = ^x 2 Lm 2 + 22,xx'Lmm' 



= Hx 2 ( Wm) 2 + 2'Zxx'Lmm . . .(14). 



