On the General Properties of Algebraic Surfaces, by the Rev, 
Dionysius Lardner, M. R. I. A. 
Read May 24, 1824. 
(1) SURFACES represented by algebraic equations have many 
properties analogous to those of algebraic curves, which have been 
investigated in the twenty-first section of the first Part of my 
Geometry. These properties are, for the most part, investigated on 
principles exactly similar to those used in that section, and it will 
be therefore unnecessary to enter with the same minuteness into 
the details of the reasoning. 
Surfaces, like plane curves, are classed according to the degrees 
of the equations which represent them, and for the same reasons. 
An equation between three variables may not always represent a 
surface of that degree marked by its dimensions. This happens 
when it is capable of being resolved into rational factors of inferior 
degrees, in which case it represents as many different surfaces of 
inferior orders. It may even represent only a system of planes, 
which takes place when it can be resolved into rational and real 
factors of the first degree. Thus the observations in (598) and 
(599) of the Geometry apply equally to surfaces represented by 
algebraic equations. 
Prop. 
(2) To determine the number of given points through which an 
algebraic surface may be subjected to pass. 
If the coordinates of the points be successively substituted in 
the equation of the surface, there should be as many equa- 
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