The Besultant of a Set of Homogeneous Lineo-Linear Equations. 379 
The performance of subtraction thus enables us to divide by r] and to 
arrive at an equation containing only the variables 'C, rf , 'C, ^-q, rji^, 
namely, the equation 
( _ 12467 + 13457)^ + ( - 12568 + 23458)^^ + ( - 13569 + 23469)^^ 
+ ( - 12468 + 13458 - 12567 + 23457)s^>7 
+ ( - 12569 + 23459 - 13568 + 23468)^7^ 
+ (- 12469 + 13459 -13567 + 23467) 4t = 0. 
Similarly it is found that 
( - 14597 + 14678)^'-' + ( - 24589 + 25678)r/^- + ( - 34689 + 35679)^ 
+ ( - 24579 + 24678 - 14589 + 15678)£,7 
+ ( - 34589 + 35678 - 24689 + 25679)^?^ 
+ ( - 34579 + 34678 - 14689 + 15679);4 = 0, 
and that 
( - 13478 + 12479)r + ( - 23578 + 12589)^f + ( - 23679 + 13689)^ 
+ ( - 13578 + 12579 - 23478 + 12489)^^r7 
+ ( - 23678 + 12689 - 23579 + 13589)/?^ 
+ ( - 13678 + 12679 - 23479 + 13489)4^ = 0. 
The final eliminant is thus a determinant of the 6th order, each of 
whose elements is either a five-line determinant or a sum of five-line 
determinants. 
9. Almost all this work, however, is unnecessary, in view of the fact 
that the original set of equations is invariant to the simultaneous circular 
substitutions 
1, 2, 3 = 2,3, 1; 
^,v,^ = ri,^,^; 4,5,6 = 5,6,4; (S,) 
7,8,9 = 8,9,7; 
and also to the simultaneous circular substitutions 
1, 4, 7 = 4, 7, 1 
u, V, iu = v, lu, u ; 2, 5, 8 = 5, 8, 2 (S^) 
3, 6, 9 = 6, 9, 3. 
Making use of this invariance, we can construct the eliminant with ease 
as soon as two elements of the first row are known. 
Thus, the first element of all being 12347, substitution gives 23158, 
31269 for the next two elements of the first row : and the fourth element 
being the sum of the first two elements with columns 7 and 8 inter- 
changed, the fifth element will be the sum of the second and third with 
columns 8 and 9 interchanged, and the sixth will be the sum of the third 
26 
