G82 DR THOMAS MUIR ON THE 



or, in the abridged notation just explained, 



(1, 0-1, + 1, + 6); 

 and 



which will be found to be equal to 



(0, 1 + 2, -1 + 2, + 0). 



Calling these two aggregates a- and a-', so that 



a— a-' = (1, -1-3, 1-1, + 6), 

 we have 



B'-2cr' = B, A + o— er' = B, C+2o— <r = 3M 8 ; 



C'-4(r' = C, B + o— </ = C, 



M' 3 -2cr' = M 3 , J+cr-cr' = M 3 , 



and consequently, if it were desired to have all the cubics expressed in terms of M s , we 



should have 



A = 3M 3 + 3o-' - 4o-, 



B = 3M 3 + 2a-' - 3cr, 



B' = 3M 3 + 4o-' - 3<7, 



C = 3M 3 + a-' - 2a; 



C = 3M 3 + 5<r' - 2(r, 



M' 3 = M 3 + 2c/, 



J = M 3 + a-' - <r. 



(23) As an illustrative example, let us again take Sylvester's second case, where, as 



we have seen, 



u x = Ax 2 + ayz + bzx + cxy, 



u 2 = My 2 + lyz+mzx+nxy, 



u s = B.z 2 +pyz + qzx+rxy. 



Following closely the rule of § 19 we write the equations in the form 



(Ax+cy)-x + az-y + bx-z = 



ny-x + (My + lz)-y + mx-z — 



ry-x + pzy + (Rz + qx)-z = 



and eliminating x, y, z we obtain 



Ax+cy 



az 



bx 



ny 



My + lz 



mx 



ry 



pz 



R.z + qx 



or 



AMq-xhj + MRc-y 2 z + ¥LAlz 2 x 



+ M(qc — br)-xy 2 + R(c£— na)-yz 2 + A(lq—pm)x 2 z + (A + AMR)^. 



