232 Proceedings of Eoyal Society of Edinburgh. [sess. 
Instead of writing the first equation in the form 
(Ax + cy)x = - (bx +• ay)z , 
we make the simple change 
(Ax + bz)x = - (ex + az)y, 
(M y + nx)y = -(ly + mx)z , 
(Rz + py)z = - (qz + ry)x ; 
whence by multiplication there results 
(Ax H- bz)(M.y + nx)(Rz +py) + (ex + az)(ly + mx)(qz + ry) — 0 , 
and similarly, 
and 
or 
(A np + cmr)x 2 y + (M pb + lra)y 2 z+ (Rbn +qam)z 2 x 
+ (MR& + alq) yz 2 + (R An + mqc)zx 2 + ( AMy> + rcl)xy 2 
+ (AMR + amr + bnp + clq)xyz — 0 . 
Instead, therefore, of the 7th row of the determinant of § 14 we 
may substitute the row 
Anp + cmr M pb + Ira R bn + qam MR& + alq RA n + mqc AMp + rcl <r + a ; 
and since the last element of the new 7th row is not itself new, it 
follows that 
AM# + bnr ) 
or that 
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(7,1) (7,2) (7,3) (7,4) (7,5) (7,6) . 
= 0 , 
