CHAPTEB XII 



GENERAL EQUATION OF THE SECOND DEGREE 

 Ax* + 2 Hjry + By* 



175. General equation of the second degree in two variables. 

 Thus far only special equations of the second dr^ivr 

 been studied ; they have all been of the form 



.(7-0, ... ,1, 



i.e., they have been free from the term containing tin- 

 product of the variables. In Arts. 107, 113, and 119 it is 

 shown that equation (1) represents a conic section having 

 its axes parallel to the coordinate axes. It still remains to 

 be shown, however, that the most general equation of the 

 second degree, vi/.. 



Ax* + 2Hxy + By* + 2Gx + 2Fy + (7=0, . . . 



also represents a conid section. To prove this it is only 

 necessary to show that, by a suitable change of the coordi- 

 nate axes, equation (2) may be reduced to the form 

 equation (1). 



If equation (2) be referred to new axes, OX 1 and oy. 

 say, making an angle with the corn^pondin^ given a \ 

 and if the new coordinates of any point on the curve 1* 

 and y', the old < -oordinates of the same point being x and y ; 

 then (Art. 72) 



x = x 1 cos y 1 . sin 0, and y = x 1 sin 6 + y' cos 6. . . (3) 



292 



