192 MR. CHARLES CHAMBERS ON THE 



XXV. On the Geometrical Representation of the Equation 

 of the Second Degree. By Charles Chambers, F.R.S., 

 Superintendent of the Colaba Observatory, Bombay. 

 Communicated by J. A. Bennion, F.R.A.S., A.C.P.S. 



Eead December 26th, 1877. 



For every value from — c» to +00 of one of the variables 

 in an equation of the second degree between two variables, 

 there are corresponding pairs of values of the other vari- 

 able j and for every value from — 00 to + cx3 of the second 

 variable there are corresponding pairs of values of the first 

 variable. The corresponding pairs of values are of two 

 classes, viz. first those which are not, and secondly those 

 which are, affected by the symbol V — i • The Cartesian 

 method gives a perfectly clear geometrical representation 

 of that part of the equation for which the values of both 

 the variables are real, but discards as unintelligible that 

 part for which either of those values is imaginary. In the 

 simplest case (of rectangular coordinates) the unit adopted 

 in the Cartesian system, as the analogue of the unit of the 

 algebraical equation, is a unit of length of a mathematical 

 straight line, with a convention as to signs which we need 

 not describe. For a reason that will presently appear, we 

 here make the otherwise simple observation that, if we sub- 

 stitute (in the Cartesian method) the idea of pairs of co- 

 incident ordinates and take the product of their values 

 instead of the square of a single ordinate, we only add a 

 convention that leaves the analogy between the algebra 

 and the geometry in question untouched. We further re- 

 mark that this analogy will still be perfect, if we conceive 



