OF THE SECOND DEGREE. 



36 



the disadvantage of introducing such a multitude of trigono- 

 metrical symbols as to give the whole discussion the appear- 

 ance of a chapter on trigonometry. 



In the method vphich I propose to substitute for this latter 

 transformation, the science is made to depend on its own 

 resources and notations, with little or no reference either to 

 the theorems or symbols of trigonometry. Another important 

 advantage of the method which I propose is, that the investi- 

 gations relative to oblique axes are very little, if at all, more 

 difficult than those which relate to rectangular axes. For 

 the sake of brevity, I have confined myself throughout the 

 paper to the most general case in which the co-ordinates are 

 oblique. 



The equation of curves of the second degree can always be 

 reduced to the form 



A3/24-2 Bi/x + Cx^ + 2 Dy + 2 Ea; + F=o (1), 



where A is a positive integer, and B, C, D, E, F may be 

 positive or negative integers. For, if the co-efficient of y^ be 

 negative, it may be made positive by changing the signs of 

 all the terms of the equation; and if the co-efficient of x, y, 

 or xy, be odd, it can be made even by multiplying the whole 

 equation by 2. 



Let R be the distance from a given point A {xi yi) to any point 

 xy on the curve (1), then by the theory of the straight line 



R'=(y— yi)' + (a;— a;i)* + 2 (a;— a^i) (i/—yi) cos y, 

 y being the angle made by the positive axes of x and y. 

 Again, let m be the direction index of the straight line R ; then 



y—y\=in {x—x^ (2), 



and in virtue of this, the preceding equation becomes 



R2=:(x— ici)' (m' + 2 w cos y+1) (a). 



Now since Ay^=A {y—y\f + 2 Ayi {y—y{) + Ayi\ 

 Cx' = C {x—xiY + 2 Cx^ (x—o^i) + Cari% 



