3G0 A. Mukliopadhyay-^MemotV on Plane Analytic Geometry. [No. 3, 



The first point of departure, then, in the classification of conies, depends 

 on the equation 



ah-h^ y ov ^ 0. 

 The case in which h^ = ah does not admit of the above transformation, 

 and it must be treated separately (see Carr's Synopsis of Pure Mathe- 

 matics, §§. 4430 — 4443). In the case where (ab — h^) does not vanish, 

 we proceed further, as follows. Turn the axes about the new origin 

 through an angle 0, where 6 is given by 



tan 2^=-^, (36) 



a — 



and the new equation becomes 



^"'+^^'-^^=*' <^^)' 



where A, B are certain constants to be determined hereafter. This 

 equation may be put into the form 



5+f.=i ^38) 



« l—% w^-% (^^)'<^<^> 



^-^ ^-^' (^i> 



Definition. — The quantity which we have denoted here by Q, we 

 will call the Asymptotic Constant, the reason for which name will 

 appear in §. 12. The quantities a, /8 are called the semi-axes of the 

 conic. 



§. 10. Invariants. — In the last section, we transformed the ge- 

 neral equation of the second degree to its simplest form (38) ; but, we 

 did not calculate the quantities a, y8 which depend on A, B. As a rule, 

 the calculation of these quantities in every particular case is a laborious 

 task ; we, therefore, find out some functions of the coefficients which 

 remain unaltered by transformation, and which are, accordingly, called 

 Invariants of the conic. These invariants may be of different classes ; 

 thus, there are certain quantities which remain unaltered for a transla- 

 tion-transformation, and which may appropriately be called Transla- 

 lation-invariantS ; to this class belong a, /i, h. Again, there are 

 certain quantities which remain unaltered for a rotation-transformation, 

 and which may, accordingly, be called Rotation-invariantS ; thus, 

 the absolute term is a rotation-invariant ; but the most important of 

 these invariants are embodied in Dr. Boole's theorems that the quanti- 

 ties 



a4-^ — 2/icosw ah — Ji^ . ^. . ^. 



--^ r-T , ■ . ' (42), (43) 



