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



It has now to be remarked that in the case of more than four 

 variables there will be relations among the coefficients of the 

 expression on the right-hand side of equation (2), the case of 

 five variables introducing a single relation. In these cases the 

 equations given above furnish us with the laws obeyed by a set 

 of non-commutative symbols which will enable us to facto rize 

 the given expression, as may be verified by the direct multiplica- 

 tion of the factors given above ; but it is only in special cases, 

 where relations exist among the coefficients, that these symbols 

 can be identified with matrices of the second order. This, how- 

 ever, will not affect the validity of our theory. We have, in fact, 

 hit upon a generalization of the theory of matrices of the second 

 order. It is easily verified that any expression of the form 



X^+p^X^-\-p^X^+ +P„-i^„> 



where the X's are ordinary algebraical quantities, satisfies an 

 equation of the second degree with scalar coefficients, so that the 

 theory of articles 2 and 3 will apply. The only difference occurs 

 in the derivation of a set of scalar solutions from the solution 

 involving non-commutative symbols. In the case of more than 

 four variables, we have more than four scalar solutions. However, 

 as the full discussion of this point would make the present com- 

 munication too lengthy, I have reserved it for another paper. 



6. It now only remains to point out the application of the 

 foregoing theory to several of the equations that occur in the 

 application of Mathematics to Physics. 



As particular examples of the case we have worked out in 

 full, we have the equations 



I have given the theory for Laplace's Equation in a former 

 communication to the Society*. To adapt the foregoing theory 

 to the case of the second equation we must write 



u = X — pt, V = y — qt, 



where the matrices p and q satisfy the equations 



p^ = a^, (f = a\ pq + qp = 0. 



Also for the linear relations connecting the differential coefficients 



* Proc. C. P. S., vii., 151—156. See also vii., 120—126. 



