680 DR THOMAS MUIR ON THE 



three of the form 



(a 1 x+2g 1 z + 2h 1 y)-x + (btf+feyy + (cf+f^yz: 



three of the form 



(a 1 x + 2ff 1 z + h 1 ij)-x + (btf+fa + hixyy + (c^+f^yz: 



three of the form 



(a 1 x+g 1 z+2h 1 y)x + {\y+f x z)y + (fif+fff +&*)*: 



three of the form 



(a 1 x+2g 1 z + h 1 y)-x + {\y-\-h x xyy + (c 1 z + 2/ 1 y)-2 : 



three of the form 



(a 1 x+g 1 z + 2h 1 y)-x + (b^ + 2/^yy + (pp+g x n)+\ 



one of the form 



(a 1 x + 2h 1 y)-x + (\y -\-2f x z)-y + (c 1 z + 2g 1 x)-z : 



one of the form 



(a 1 x + 2h 1 y)-x + (b^ + 2/^yy + (c 1 z + 2g 1 x)-z: 



and one of the form 



(ajX + htf+gjzyx + Ww+fp+hfify + (c 1 2+^ 1 x+/ 1 y)-2. 



Each of the last three is unique, by reason of being invariant to the cyclical change. 

 It is the last of all which gives rise to the Jacobian ; and this, strange to say, is the 

 lengthiest cubic in the whole series, as many as 17 of the 20 determinants of § 12 

 appearing in it. 



The second from the end gives rise to M 3 . The third from the end originates a 

 cubic equally simple with M 3 ; so that, exactly as in the case of the eliminant of the 

 6th order, we have two alternative forms. On turning to the rule of § 19, it will be seen 

 that from the three parts of each of the given quadrics we excluded z, x, y respectively 

 when we might equally reasonably have excluded y, z, x: it is this exclusion of y, z, x 

 from the three parts respectively which gives rise to the alternative form. 



(21) Out of the 18 partitions there arise only 8 cubics which are invariant to the 

 cyclical change ; for, in five different instances, it is necessary to use three of the parti- 

 tions to secure one such result. 



Thus the first partition 



(a 1 x+2g i z+2h 1 y)x + (b$+f x »yy + (o^+f-Jf}* 

 gives the irregular cubic 



- 2[2]y - 2[3]- 2 s + {2[5]-2[5']}-2/% + [6].sftc 

 + VQ*& + {2[8] + 2[8']}.^ 2 

 + [0]-seyz; 



