619 
629] ON THE LINEAR TRANSFORMATION OF THE INTEGRAL 
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To prove in the case of a convex quadrilateral that (AO) is not = + {BD), it is 
sufficient to consider 
where A, B, C, D are the points (1, 0), 
(0, 1), (— 1, 0), and (0, — 1) respectively, and where, writing v = iu, it at once appears 
that we have 
that is, 
(AC) = ± i (BD), not (AC) = ± (BD). 
But I consider the general question of the linear transformation. If a', b', c', d' 
correspond homographically to a, b, c, d, then to represent these values a, b', c', d' we 
have the points A', B', C', D', connected with A, B, C, D according to the circular 
relation of Mobius; and then, making u, a, b', c', d' correspond homographically to 
u, a, b, c, d, and representing in like manner the variables u, v! by the points U, U' 
respectively, we have the circular relation between the two systems U, A, B, C, D 
and U', A', B', C', D'. 
Before going further I remark that the distinction of the convex and reentrant 
cases is not an invariable one; the figures are transformable the one into the other. 
Thus, taking C on the line BD (that is, between B and D, not on the line produced), 
there is not this relation between B', C', D', and the figure A'B'C'D' is convex or 
reentrant as the case may be. Giving to C an infinitesimal displacement to the one 
side or the other of the line BD, we have in the one case a convex figure, in the 
other case a reentrant figure ABCD; but the corresponding displacement of C r being 
infinitesimal, the figure A'B'C'D' remains for either displacement, convex or reentrant, 
as it originally was; that is, we have a convex figure ABCD and a reentrant figure 
ABCD, each corresponding to the figure A'B'C'D' (which is convex, or else reentrant, 
as the case may be). 
Writing for convenience 
a, b, c, f, g, h =b — c, c — a, a —b, a — d, b — d, c — d, 
a', b', c', f', g', W = b' — c, c' — a', a' — b', a'— d', b' — d', c' — d', 
so that identically 
af + bg + ch = 0, a'f ' + b'g' + c'h' = 0, 
then the homographie relation between (a, b, c, d), (a', b', c', d') may be written in the 
forms 
af : bg : ch = a'f' : b'g' : c'h', 
or, what is the same thing, there exists a quantity N such that