210 Mr Brill, On Linear Differential Equations. [April 80, 



Proceeding to the case of four variables, we have the equation 



In this case we shall have for our matrical solution a function 

 of the form 



A +tiB + vG + wD 



+ gT {u^E + v'F + w^G + (vw + wv) H + {vm + uw) K + (uv + vu) L] 



+ H-, [u^M + v^'N + vfP + {v'w + vwv + wv") Q + &c.} + &c., 



where u = £c—pt, v = y — qt, w — z~rt, 



p, q, r being matrices satisfying the equations 



p^ = q^ = r^ = (x^ 



qr + rq — rp + pr =pq + qp = 0. 



By means of the theory as applied to four variables we may 

 also discuss the equation 



In this case we must write 



u = y —px, v = t — qx, w = 1 —rx, 

 where p, q, r satisfy the equations 



p' = -l, q''^r' = 0, 



1 



qr + rq — —^, rp + pr = pq + qp = 0. 



Finally we have the equation 



In this case we shall have for our solution a function of the 

 above type constructed with the four compound variables 



y — px, z — qx, t — rx, 1 — sx, 



where the symbols p, q, r, s obey the laws expressed by the 

 equations 



p^ = q^^-l, r^^s' = 0, 



pq + qp =pr + rp =ps + sp = qr + rq = qs + sq = 0, 



1 



rs + sr — --^. 

 a 



In this case our non -commutative symbols cannot be identified 

 with matrices. 



