156] 
A FIFTH MEMOIR UPON QUANTICS. 
535 
the bipartite quadric (x — ay) (x — ay), yet the theory of involution cannot be completely 
discussed except in connexion with that of homography. If we write 
A = (0 - 7) (a - 8), B = (y — a) (0 -8), G = (a - 0) (7 - 8), 
A' = (0'-y')(a'-8'), B' = (ry'-a')(0'-8'), C' = (a'-0')( 7 '- 8'), 
then we have 
A + B 4- C = 0, 
A' + B'+C' = 0, 
and thence 
BC' - B'C = CA' — G'A = AB'- A'B: 
and either of these expressions is in fact equal to the last-mentioned determinant, as 
may be easily verified. Hence, when the determinant vanishes, we have 
A : B : C= A' : B' : C\ 
Any one of the three ratios A : B : C, for instance the ratio B : C,= 
(7-«) (/3-8) 
(a - 0) (y -8)’ 
is said to be the anharmonic ratio of the set (a, /3, y, 8), and consequently the two 
sets (a, A, y, 8) and (a\ 0', y', 8') will be homographically related when the anharmonic 
ratios (that is, the corresponding anharmonic ratios) of the two sets are equal. 
If any one of the anharmonic ratios be equal to unity, then the four terms of 
the set taken in a proper manner in pairs, will be harmonics; thus the equation 
B . . 
£ = 1 gives 
(y - a ) (/3 - S) 
(«-/8) (y-8) 
= 1, 
which is reducible to 
2a8 + 2/3y — (a + 8) (¡3 + y) = 0, 
which expresses that the pairs a, 8 and 0, y are harmonics. 
109. Now returning to the theory of involution (and for greater convenience 
taking a, a! &c. instead of a, ¡3 &c. to represent the terms of the same pair), the 
pairs a, a!; 0, ¡3'; y, y'; 8, 8'; &c. will be in involution if each of the determinants 
formed with any three lines of the matrix 
1, a + a', aa! , 
1, ¡3 + ff, 00', 
3, y + yri yy', 
1, 8+8', 88', 
&c.