538 
[547 
547. 
ON THE REPRESENTATION OF A SPHERICAL OR OTHER SUR 
FACE ON A PLANE: A SMITH’S PRIZE DISSERTATION. 
[From the Messenger of Mathematics, vol. n. (1873), pp. 36, 37.] 
In the Smith’s Prize Examination for 1871 I set as the subject for a dissertation: 
The representation of a spherical or other surface on a plane. 
I give the following as a specimen of the sort of answer required: an answer 
which, without so much as noticing that projection (in its restricted sense) is only one 
kind of representation, goes into the details of the constructions for the different 
projections of the sphere, and even into the demonstrations of these constructions, errs 
quite as much by excess as by defect, and is worth very little indeed. 
The question is understood to refer to Chartography, viz. the kind of represen 
tation is taken to be such as that of a hemisphere or other portion of the earth’s 
surface in a map. 
An implied condition is that each point of the surface (viz. of the portion thereof 
comprised in the map) shall be represented by a single point on the map; and con 
versely, that each point on the map shall represent a single point on the surface. 
And further, any closed curve on the surface must be represented by a closed curve 
on the map, and the points within the one by the points within the other. If for 
shortness the term element is used to denote an infinitesimal area included within a 
closed curve, we may say that each element of the surface must be represented by an 
element of the map; and conversely, each element of the map must represent an 
element of the surface.