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



Therefore, we have 



d ./(m) = dm . 1^/' (\) + '^r (\) 

 [\ — \ \ — \ 



= dm . f (m), 



where /' (m) is the matrical function framed on the model of the 

 scalar function f (x). 



Supposing m to vary only as depending on the three scalar 

 variables so, y, z, we may write the above equation in the form 



This being true for all possible values of the ratios dx:dy: dz, 

 we see that the equation resolves itself into the three 



Thus if we write 



^ dec ^ dy dz ' 

 we have ^fi'^n) = Am .f (m). 



Hence if m be such that 



%n — (X, +\) 1 A r> 



^^- — -^ = const., and Am = 0, 



then we have also A/(m) = 0. 



3. If we write 



u = y — px, v = z — qx, 



then we have Au = 0, and Av = 0. Further, writing m= ^u + rjv, 

 where | and 77 are scalar constants, we have Am = 0. 



If \ and \ be the latent roots of m, we have 



tv 



00 



W = ib + vzf - ;, (?2/ + vz) {h^ + gv) 



