568 A SIXTH MEMOIR UPON QUANTICS. [158 
harmonic ratios of the tetrads of the first system are homographically related to the 
anharmonic ratios of the tetrads of the second system. And it is not material which 
of the three anharmonic ratios of a tetrad of either system is selected, provided that 
the same selection is made for each of the other tetrads of the same system. The 
order of the points of a tetrad must be attended to, but there are in all four 
admissible permutations of the points of a tetrad, viz. if- A, B, G, D are the points 
of a tetrad, then (A, B, C, D), (B, A, D, C), (C, D, A, B), (D, C, B, A) may be 
considered as one and the same tetrad. Any three tetrads whatever in the second 
system may correspond to any three tetrads of the first system; and then given a 
fourth tetrad of the first system, and three out of the four points of the corre 
sponding tetrad of the second system, the remaining point of the tetrad may be 
determined. The words line and point need not, as regards the two systems of 
tetrads respectively, be understood in the same significations.—See Fifth Memoir, 
No. 112. 
165. The foregoing theory of the harmonic relation shows that if we have a 
point-pair 
(a, b, c$x, y) 2 = 0, 
the equation of any other point-pair whatever can be expressed, and that in two 
different ways, in the form 
(a, b, c§x, y) 2 -f (lx + my) 2 = 0 ; 
the points (lx + my = 0) corresponding to the two admissible values of the linear 
function being in fact the harmonics of the point-pair in respect to the given point- 
pair (a, b, c§x, y) 2 = 0, or what is the same thing, the sibiconjugate points of the 
involution formed by the two point-pairs (see Fifth Memoir, No. 105). The point-pair 
represented by the equation in question does not in itself stand in any peculiar 
relation to the given point-pair (a, b, c§x, y) 2 = 0; but when thus represented it is 
said to be inscribed in the given point-pair, and the point lx + my = 0 is said to be 
the axis of inscription. And the harmonic of this point with respect to the given 
point-pair (that is, the other sibiconjugate point of the involution of the two point- 
pairs) is said to be the centre of inscription 1 . 
166. We may, if we please, (x', y') and 6 being constants, exhibit the equation 
of the inscribed point-pair in the form 
(a, b, c$x, y) 2 (a, b, c§x', y) 2 sin 2 6 — (ac — b 2 ) (xy' — x'y) 2 = 0, 
where we have for the axis of inscription and centre of inscription respectively, the 
equations 
xy' - x'y = 0, 
(a, b, C&X, y$af, y') = 0] 
1 The words inscribed, inscription, are used not in opposition to, but as identical with, the words cir 
cumscribed, circumscription; and in like manner (post, Nos. 203 et seq.) as regards conics.