The collected mathematical papers of Arthur Cayley.

234

A SMITH'S PRIZE PAPER AND DISSERTATION;

L591

and hence
Tn- /d 7d 7 v cdu dv 7
J2 (hk/-h1k) =(dT - d )8v, = J8uv v,
x= dx dy dy dx x 8uv, =JSu)v,
that is,
hk/ - hk = y 818v,
and 8u, 3v being as above each of them positive, J has the same sign as hki -hlck
But the rotation from OX' to O Y' is in the same direction as that from Ox to Oy,
or in the contrary direction, according as hk/ - hzl is + or -, that is, according as
d dudv du dv
J, = - d d is + or -; which is the theorem in question.
dox dy dy dx'
13. Write a dissertation on:
The theory and constructions of Perspective.
In Perspective we represent an object in space by means of its central projection
upon a plane: viz. any point P,1 of the object is represented by P', the intersection
with the plane of projection of the line DP, from the centre of projection (or say
the eye) D1 to the point P1; and considering any line or curve in the object, this is
represented by the line or curve which is the locus of the points P', the projections
of the corresponding points P1 of the line or the curve in the object.
The fundamental construction in perspective is derived from the following considerations: viz. considering through P1 (fig. 1) a line meeting the plane of projection
in Q, and drawing parallel thereto through D1 a line to meet the plane of projection
in M and joining the points M, Q, then the lines DM 1, Q, QP are in a plane;
that is, the plane through D1 and the line P1Q meets the plane of projection in MQ;
Fig. 1.
and consequently the projection P' of any point P1 in the line P1Q lies in the line
QM; and not only so, but considering only the points P1 of this line which lie
behind the plane of projection (D1 being considered as in front of it), the projections
of all these points lie on the terminated line MQ; viz. Q is the projection of the
point Q, and M the projection of the point at infinity on the line QPI; or, if we
please, the finite line QM is the projection of the line QPloo.
* The subscript unity is used to denote a point not in the plane of projection, considered as a point
out of this plane; a point in the plane of projection, used in the constructions of perspective as a conventional representation of a point P,, will be denoted by the same letter without the subscript unity.
And the like as regards Di and D.