ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 391 
—a form more suitable for obtaining the cofactors of the elements of the last row in 
terms of the familiar three-line determinants 0,1, 2,.... Taking first the cofactor 
of 0+0’, and using Lapracr’s expansion-theorem, we find it 
0(00 —16) —7(-89’) + 4(02—17')- 3(05+69’) + 9(78)-8'(25-7'9), 
= 000-— 016 — 5147’ 4+ 2°7894+ 789’. 
Similarly the cofactor of 4 is found to be 
— 002 + 038 + 056 +03'6— 189 —1'48+4 224 
— 255’ +347’ — 4'99' 559 +5'7'9’ + 669° + 6’89, 
and the cofactor of 4’ to be 
—004+ 013 +036'+ 067 —119 —11'44 1’8’9’ 
+ 944 + 268’ — 257 — 2'99' + 339’ 4 3/79 —579. 
The full eliminant is thus 
(0 + 0’)[000 - 3016 — 5147’ + 2-789 + 7'8'9’] 
+ S'47 002 + 038 +036 +056 — 1/48 — 189 an 
| — 255’ + 3/47’ — 4'99’ — 559 +. 5'7’9’ + 669’ + 689 
+ 2" — 004+ 013 +036'+ 067 —119 —11'44+ re 
+ 244 4+ 26'8' — 25/7 — 2'99’ + 339’ 4+ 379 — 579 |. 
This does not, of course, differ from the form used in § 41. Asa matter of fact, it will 
be found on examination that nineteen of its thirty-eight terms agree with terms in the 
expression of § 41, and that the other nineteen can be changed without much diftculty 
so as to establish the identity of the two expressions. 
(46) As may be supposed, however, the importance of the new result does not 
consist in its affording a verification of that previously obtained. It is more interesting, 
in fact, in its unsimplified state; for it has now to be noted that each of the three 
lengthy expressions found in it as the cofactors of 0 +0’, 4, 4’ can be put in the form of 
a simple three-line determinant. For example, 
58" 59 | 
ta) = 
4 
9 
47’ 
BEN; GO(, 47) . 67 48 59 : 69\59 
vA Yoetoutl) « B.. Ho ne Silent 
(047 ep cr ee ee Oe 
58’ 69" 47’ 7/9 587 698’ O59) po 6 99) 
000 +.00(2° alll 0( l ) 8/9’ 4 2-789 — 59 _ $7699 
+00 +a) + pe OE) 4 ree + 2789 — D107 > 
000+ 3105(S =) + S69 = ) 4789 +2789, 
000 — 053 — 5369’ + 7'8'9'+2-789, 
000 — $016 — 147’ + 7'8'9'+2°789, 
rs 
(o2) 
Il 
= cofactor of 0+ 0’, 
