ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 27 



but unfortunately in this case there is no simple equivalent which can be substituted for 

 either. This difficulty, however, can be overcome by taking one of the terms common 

 to the two original expansions, viz. ±1268, and adding it to each side, for then we have 



-±16711 + ±1788' + ±1268 = + ±16711 - ±155'6 + ±1268, 



= -±16711 + ±16711. 



It is thus seen that in each of the original expansions an aggregate of thirteen terms 

 may be supplanted by an aggregate of seven, viz. 



±04911 - ±471112 + ±44611 - ±77812 + ±16711 - ±16711*+ ±16711*; 



and that it only remains to prove the equality of 



07'8'9' - 0'456 and -04W - 0789 , 



and if possible to find for either of them a simpler equivalent. 

 Beginning with the left-hand side we derive from 



07' = 45 - 810 , 0'6 = 8'9' + 34', 



in the same manner as before the identities 



±07'8'9' = ±458'9' - ±88'9'T0 , 

 -±0'456 = -±458'9' - ±344'5 , 



* For each of the terms 2l67'll, 2l6'711 an alternative form is available, by reason of the existence of a curious 

 kind of identity of which there are three instances, viz. : — 



2l67'TT = 2l59'TT, 



o o 



2i6'7iT = 2i48'iT, 



S44'88' = 24'589' . 



The mode of establishing these may be illustrated by proving the last of the three. 

 By a well-known theorem we have 



I a AU I I c i% 3 1 = ! s'A/s i I <hK a s I + i c hy-ifs ! I c i h A I + I a i & 2</ 3 1 I c A/ 3 1 . 



i.e., 76' = 5'9 - 9'5 + 48', 



or ' 76' - 95' = 48' - 59', 



where, be it observed, each side consists of two terms of a triad. Multiplying, then, both sides by the remaining term 

 of either triad, say by 84', we have 



84'(76'-95') = 84'(48' - 59') , 

 and therefore by cyclical substitution 



95'(84'-76') = 95'(59' - 67') , 

 and 76'(95'-84') = 76'(67' - 48') . 

 From these by addition there results 



= 284'48' - E84'59' 

 or 24'589' = 2°44'88'. 



The three fundamental identities which can be treated in this manner are 



76' - 95' = 48' - 59' = 1 11 - 2 12, 

 or, of course, their derivatives by cyclical substitution. 



