396 DR THOMAS MUIR ON THE 
From these we obtain 
be 1 ey 7 
(0+ 5) 5 oe? + Txy? + (4-5 Vays = 0 
5 ohe 
8y22 + (0+ 38" )eu? o iy? + (5- 3 2! nye = 0 
3 Be es ae ~ 
7 Bye? + 92a? + Oa aye Oe 2 
The performance of the interchange of § 42 gives the companion set 
(0+ 7 8')a2y + By22 + 5 922m + (2- 5")xye = 0 
4 4 4 
6 6 , 2 ~2. 6 U ra 
gialy + 0+ 59 ye + 922” + 3= 56 eye —. 0 
Ta2y + 4 By + (04 57’) + (a- 5 eve a) 
The necessary fourth equation for the cases alluded to in the preceding paragraph 
is got from the cubic M, of § 9, 
4aPy + yz + 622a + 4 yz? + 52x? + 6’ xy? + (0+0')xyz = 0, 
or from its companion M’, , 
ll 
S 
ary + 3'y2t Vax + Qy2? + 320? + Lay? + (0+ 0')ayz 
by putting three appropriate coetticients equal to zero. 
(52) Another special case of similar type is still more interesting, viz., the case where 
7’, 8’, 9’ vanish. The Jacobian of the given set of equations, viz. 
—D(8'a*) + S(244+2')\e2y + D224 4’ )ye? + (404+ 0')ayz = 0, 
then loses three of its terms; and as the operation 2M, + M’, gives 
D244 2')ary + D(2-4'+2)y24+3(04+0')ayz = 0, 
it 1s clear that there follows 
2(2-4 )yz?+(0-2°0')ayz = 0, 
—an equation which can be used to complete the first set of three in § 51. Since the 
vanishing of 7’, 8’, 9’ makes 11’, 22’, 33’ identical, the resulting eliminant is 
0 — 7 4-2! 
2 
27 On 
8 a - ; 
3 5-3 
= 9 0 6-1 
2-4’ 3-5' 1-6’ 0-20’ 
