202 Mr Brill, The Application of Matrices to the [April 30, 



by its equivalent 



d 



a^r- 



ox 





Any matrical function of x, y, z, which is made to vanish by 

 either of the two linear operators, may be considered as a matrical 

 solution of the equation 



a^ a^ 3^ d'e d'e 3^6' _ 



dx' dy^ dz^ '' dydz ^ dzdx dxdy~ 



2. Let Xj, \ be the latent roots of the matrix m, then the 

 identical equation satisfied by the matrix may bo expressed in 

 the form 



m^ - m {X^ + \) + W = 0. 



Differentiating, we have 

 m . dm + dm.m- dm (\ + \) — m {d\ + d\^ + \d\ + \dX^ = 0. 



We will now assume 



, Qii — A.„ - m — \,, , 

 dm = ^ -? dX, + r -^ dX^ + dco. 



\-\ \-\ 



Substituting this value for dm in the above equation, we 

 obtain 



(2m -X^- XJ \^ -"dX^ + -^ dXS + (m - \ - \) dto 



{K -\ K-\ ) 



+ do) . m — (m — Xg) (iX^ — (m — Xj d\ = 0, 

 which reduces to 



fj-\ ^\ 



2 (m - \){m — Xg) -— ^ — r — - + (m — X^ — X ) dco + dw . m = 0. 



The first term of this equation vanishes by virtue of the 

 identical equation satisfied by m, and consequently the equation 

 reduces to 



(m — X^ — Xg) dco + d(o . m = 0, 

 which may also be written in the form 



on . dco . m = XX„d(o. 



12" 



It is not necessary for our present purposes to find the general 

 value of dca which satisfies this equation, it will be sufficient to 

 note that a particular solution is dco = 0. 



Making this assumption, which is equivalent to assuming that 

 dm shall be commutative with m, we have 



, m — X„ ,^ m — X. - ,, , 



d7n = - r^d\ + - --'d\^ (1), 



X, — X„ X„ — X, 



