532 
A FIFTH MEMOIR UPON QUANTICS. 
[156 
102. We have identically 
2a/3 + 2a 7 /3 7 -(a + /3) (a 7 + /3 7 ) 
= 2 (a - a 7 ) (a — /3') — (a - /3) (2a - a' - /S') 
= 2(0 -a')(/3 -0O-(0 -«)(20 -a'-0') 
= 2 (a 7 — a ) (a 7 — /3 ) — (a 7 — /3') (2a 7 — a — 0 ) 
= 2 (/3' — a) (/3 7 — /3 ) — (/3 7 — a 7 ) (2/3 7 — a -0); 
and the equation Q = ac' — 256' + ca 7 = 0 may consequently be written in the several 
forms 
1 
2 
1 
+ 
a — /3 a - a 7 ^ a - /3 7 ’ 
2 1 1 
/3 — a /3 — a 7 /S 7 — /3 ’ 
2 11 
IJi ~ 77 .. + 77— 
a 7 — /3' a 7 -a a 7 - /3 ’ 
2 1 1 
a/ o/ „ d - /3/ n) 
/3' - a 7 0' - a T /3' - /3 ’ 
so that the roots (a, 0), (a 7 , /3') are harmonically related to each. other, and hence the 
notion of the harmonic relation of the two quadrics. 
103. In the case where the two quadrics have a common root a = a 7 , 
a- 1 U = (x - ay) {x - /3 y), 
a 7 - 1 TT ={x- ay) (x - ¡3'y), 
4 a~ 2 □ = — (a — /3) 2 , 
2{aa)-'Q = (a -0) (a - /3'), 
4a 7 - 2 □ 7 = — (a — /3 7 ) 2 , 
E =0, 
(aa 7 ) -1 H =(/3' — /3) (a; — ay) 2 . 
104. In the case of three quadrics, of the expressions which are or might be 
considered, it will be sufficient to mention 
(1) 
(2) 
(3) 
(4) 
a', b', c' 
a", b", c 77