1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 299 



§§. 8 — 15. The General Equation of the Second Degree. 

 §. 8. Preliminary. — The discussion of the general equation of 

 the second degree deservedly occupies an important position in the 

 application of analytical geometry to the theory of lines of the second 

 order ; for, in analytical geometry properly so called, the question of 

 degree or class is of fundamental importance, and the curves of the 

 second degree should be called lines of the second order, and not conic 

 sections, the proper point of view from which their properties ought to 

 be studied being the fact that the equation representing them is of the 

 second degree, and not the other fact that they are sections of a cone 

 and have foci and directrices. The truly logical order of treating the 

 subject is first to have a chapter on the equation of the first degree, 

 containing the properties of right lines, then a chapter on the general 

 equation of the second degree, and, as distinctly subsidiary to this, 

 chapters on the circle, the ellipse, and the other conies. We proceed, 

 then, to give the barest outline of such a systematic discussion as is 

 indicated here. It may usefully be noted that the object of the dis- 

 cussion is twofold, viz., in the first place, the problem is how to trans- 

 form the equation to its simplest forms, and thus to classify the 

 different kinds of conies ; in the second place, we obtain some general 

 formulae for such properties as are common to all conies. 



§. 9. Transformation of the Equation.— -The general equation 



of the second degree being 



S=::ax'^-{-2hxy^hy^-{-2gx^2fy-\'C = 0, (32) 



first change the origin to {x', y'), so that the equation becomes 



ace^-\-2hxy-\-by'+2g'x-{-2fy-[-c'=zO 

 where 





c = Point-function. 

 If, then, we make g' =/' = 0, that is, if we have for the coordinates of 

 the new origin 



" "aZT^^' y -ab-:^-' (^^^' ^^^) 



the transformed equation is 



ax^^2hxy^hy^-^-^^^ = (35) 



where A is the discriminant (§. 4). In order that this transformation 

 may be real and possible, we must have (a& — h'^) different from zero. 



