ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 389 
and therefore in the case of products of four 
(Cay era eae 
GO eterno 1, Dir SOW gee oa, nO oils ? 
— et alae pea 
ThaaN6), 251524 29) 16", 151.4) SL 
and, lastly, in the same circumstances 
ig h as De Digaeaaii etl ss 2’, 3, » 
a y) SG 2 Gi ace 
(43) A comparison of the three interchanges, which, in the case of a four-factor 
product, we have thus found to be equivalent to 
a f b g c oh 
b AG ¢ ay te *) 
respectively, leads at once to the further observation that if the expression in which 
the interchanges have to be made be invariant to the cyclical substitution, the three 
interchanges are not essentially different. So far, therefore, as the above eliminant 1s 
concerned, we need only consider one of the interchanges, say the interchange 
Tee 2ey oh Bes all OF 
4,6,5,9; 4, 6, 5’ 
it bemg borne in mind that this implies that the determinants 
0, 730, 7 
are invariant to the interchange. The determinants 0, 0’, which are invariant to the 
eyclical substitution as well, we shall therefore speak of as being doubly-invariant. 
(44) Turning then to the elimimant and applying this interchange to each of its 
twenty-eight parts, we find that twelve of them, viz., the 
Ist, 2nd, 3rd, 4th, 5th, 7th, 11th, 15th, 18th, 21st, 26th, 27th 
are doubly-invariant ; that twelve others may be grouped as six binomials which are 
doubly-invariant, either term of each binomial being produced from the other term, viz. 
6th and 8th, 
13th and 14th, 
16th and 22nd 
I7th and 23rd, 
19th and 20th, 
25th and 28th; 
and that the four remaining parts (the 9th, 10th, 12th, 24th) are 
? 
Dogs: 0456’, -0'456, - 24468. 
Now we can show (see § 39) that 04’5’6’— 0'456 is expressible as the difference of two 
terms which are each doubly-invariant, viz., the difference 07’8’9’—0’789. Further, since 
Sb — = 119 > 1246 
