366 Relations between the Coaxial Minors of a Determinant, 

 (ab), (abc), &c, let 



2A= (bed) +2bcd -b(cd) -c{bd) -d(bc) 

 2B = (cad) -\-2cad — c(ad) r — a{cd) — d(ca) 

 2C =(abd) + 2abd-a(bd) -b(ad) -d(ab) 

 2D = (abc) +2abc-a(bc) —b(ca) -c(ab) 

 ~P = bc —(be) Q=ca—(ca) R=ab—(ab) 

 ~L = ad- {ad) M = bd- {bd) N = cd- (cd) 

 k = Aa + Bb + Cc + ~Dd — abcd 



-i { A + (be) (ad) + (ca) (bd) + (ab) (cd) \, 



so that A, B, C, D, P, Q, R, L, M, N, k are all functions of 

 coaxials. Then we find that 



ff = Y gg' = Q M'=R 



xx'-h y/=M zs'=N 



fy'z + fyz< = 2A 



gz'x + ^a/ = 2B 



Aafy + li'xy' = 2C 



#A +/VA / = 2D 

 ghyz' +g'h r y f z + h/zx' + h'fz'x +fgxy' +fg'x'y = 2k. 



In accordance with MacMahon's theorem it must be pos- 

 sible to eliminate in two different ways the twelve quantities 

 /, g, h, f g' h', x, y, z, x' y f z' from the eleven equations last 

 written and so obtain two equations connecting the coaxials. 



9. In order to effect this elimination let 



fyz' > g'zx'~ ' h'xy'~ ' 



then we find that 



A=*/MNPcosa, B=^;NLQcos/3, C=^LMRcos7, 



D='v/PQRcos(a + /3 + 7), 

 £=n/QRMNcos (/3+ 7 ) + ^RPNLcos(7 + a) 



+ * / PQLMcos( a + /3), 



and so the elimination to be performed is practically the same 

 as before. The result is seen to be 



