34 DR THOMAS MUTR ON THE 



Lastly by the elimination of x 2 , y 2 , xy and subsequent division by z we obtain the 

 equation 



{610-04-51}^+ {(8 + 8')l6-07-4'5}?/+ {-310-00 + 59}* = 0, 



and thence the eliminant 



04+ 15-616 07+ 29-810 00 + 310- 59 



00 + 111- 67 05+ 26-411 08+ 37-9IT 



09+ 18-712 00 + 212- 48 06+ 34-512 



(y 5 ) 



(38) Each of these five determinants y x , y. 2 , y 3 , y 4 , y 5 is of the 18th degree in the 

 coefficients of the original quadrics, and must, therefore, contain an extraneous factor of 

 the 6th degree. It will be seen that the first and last are essentially the same, the 

 coefficients of the equations connecting yz, zx, xy being the same as the coefficients of 

 one of the sets of equations connecting x, y, z ; that the third and fourth are more com- 

 plicated ; and that the second is still more so. 



At the outset the separation of the extraneous factor seemed likely to be a matter of 

 considerable difficulty ; a method, however, was fortunately hit upon which effects it 

 in every case with comparative ease. This will be fully understood from the following 

 application of it to the case of y x or y 5 : — 



Looking at any column of y 5 we observe that the first terms of the three trinomials 

 composing the column have a factor in common, that the second terms have also a 

 common factor, but that the same cannot be said of the third terms. In the case of the 

 first column, for example, the three third terms are —610, —67, —712, where analogy 

 would have led us to expect either a 7 in the first term or a 6 in the last. This difficulty, 

 however, can be overcome by writing 



either 79 + 04' for 610 

 or 46 - 09' for 712 . 



Taking the latter alternative the determinant becomes 



00 +111- 67 05 + 26-411 0(8 + 8')+ 37- 56 



0(9 + 9')+ 18- 64 00 +212- 48 06 + 34-512 



04 + 15-610 0(7 + 7')+ 29- 45 00 +3T0- 59 



and, as a consequence, when we proceed to express it as an aggregate of 27 determinants 

 with monomial elements, each of the 27 can have a factor removed from each of its 

 columns. Further, when the said factors have been removed, 9 of the 27 must have 

 two of their columns alike, and may therefore be neglected. Of the remaining 18 it 

 will be found that 3 are symmetrical with respect to the cyclical substitution, and that 

 15 can be grouped in triads. The following condensed form of the expansion is thus 

 readily obtainable : — 



