569 
158] 
A SIXTH MEMOIR UPON QUANTICS. 
or in the equivalent form, 
(a, b, c§x, yY(a, b, c$x', yf cos 2 6 — {{a, b, c§x, yifx, 3/')} 2 = 0, 
where we have for the axis of inscription and the centre of inscription respectively, 
the equations 
0, b, cfx, y^af, y') = 0, 
xy' — x'y — 0. 
167. The equivalence of the two forms depends on the identical equation 
(a, b, cfx, yf (a, b, c$V, yj-{(a, b, cjcc, y\x’, y')f = (ac - fr) {xy' - x'yf, 
which is in fact the equation mentioned, Fifth Memoir, No. 95. If, for shortness, 
we write 
(a, b, c\x, y) 2 = 00, 
(a, b, cifx, yjx', y') = 01 = 10, 
&c., 
then the equation may be represented in the form 
00, 01 
>3s 
1 
II 
x, y 
10, 11 
X, y' 
168. There is a like equation for the three sets (x, y), (x', y'), (%", y"); the 
right-hand side here vanishes, for there are not columns enough to form therewith 
a determinant, and the equation is 
= 0, 
00, 
01, 
02 
3 0, 
11, 
12 
20, 
21, 
22 
an equation which may also be written in the form 
cos- 
01 
Voo Vll 
— + cos- 
12 
Vll V22 
= cos -1 
02 
Voo V22 ’ 
as it is easy to verify by reducing this equation to an algebraical form. The various 
formulae have been given in relation to the establishment of the notion of distance 
in the geometry of one dimension, but it will be convenient to defer the consideration 
of this theory so as to discuss it in connexion with geometry of two dimensions. 
On Geometry of Two Dimensions, Nos. 169 to 208. 
169. In geometry of two dimensions we have the plane as a space or locus in 
quo, which is considered under two distinct aspects, viz. as made up of points, and 
as made up of lines. The several points of the plane are determined by means of 
c. 11. 72