Dawson — On the Properties of a System of Ternary Quadrics. 151 



Hence 



'T = [abc - d(hc + ca + ah) ] ~ - 4,abcd {a + h + c) d 



= a-b^c^ + a^h-d- + 5V<i^ + c^a'-d- - 2ahcd (ah + ac + ad + hc + ca + ah) ; 



Q is to be found by means of the identity 



n = 8SQ-6TP. 



We find D directly by calculating afresh for this special form the cubic which 

 is annihilated by the polar conies of 



ax^ + hy^ + cz^ - d (x + y + zy. 

 These polar conies are 



ax' - d{x + y -\- zy, by'^ - d {x + y + zf, cz^ - d [x ■¥ y + z)-. 

 Eeplace x, y, z in them by 



A A 1 



dx' dy' dz' _ 



and operate on px^ + qy^ + rz^ + '^'p^x-y + &c. + 'osxyz, 



we obtain on equating to zero the coefficients of x,y,z; and denoting by 

 /, m, n the three expressions 



^ + S'l + ^'i + 2s + 2|?3 + 2p2, ^3+ $' + r2 + 2$3 + 2s + 22'i, ^3 + 2'3 + ^' + 2^2 + 2ri + 25, 



the equations 



pa - Id = 0, hqi - Id = 0, ViC - Id = 0, 



p^a - md = 0, hq - md = 0, r^c - md = 0, 

 p-.^a - nd = 0, hqs - nd = 0, re - nd = ; 



substituting from these equations in the three 



I = 2^ + qi + r^ + 2s + 2p3 + 2p2, 



m = 2h + q + r-i + 22'3 + 2s + 2qi, 



n = 2h + §'3 + r + 2^2 + 2ri + 2s, 



Hence 



where 



I {6-1) + — (m + n) + 2s = 0, 



2d 



m (6 - 1) + 1- (n + /) + 2s = 0, 



9/7 



n(e --{) + — {I -:- m)f-2s=0, 



= di- + -+ - ., 

 .a c J 



