1887.] Plane Analytic Geometry. 233 



of the intercepted portion of the line, as well as the product of the 

 two sides, is easily found ; as an application of the formulas in this 

 section, which are all very compactly expressed in the determinant 

 notation, the area of the parallelogram formed by two lines given by 

 the general equation and two others drawn parallel to them through 

 the origin, is found. In the two following sections, some properties 

 of the circle are discussed ; the sixth section shews that the constant 

 term in the equation of a circle represents the square of the tangent 

 drawn from the origin to the circle, whence flow some interesting 

 properties ; the seventh section treats of the chords and tangents of 

 circles and conies ; the geometric meaning of Professor Burnside's 

 equation is pointed out, and the equation of the tangents, drawn from 

 any point to a conic, is obtained by a process of transformation. The 

 next eight sections contain a systematic discussion of the general 

 equation of the second degree, supplementary to what is given in 

 ordinary text-books. The eighth section contains some preliminary 

 remarks ; the ninth section treats of the transformation of the general 

 equation, and introduces the subject of the classification of conies, 

 which is completed in the eleventh section ; the term Asymptotic Con- 

 stant is here introduced and explained. The tenth section gives an 

 elaborate discussion of the invariants and covariants of a single conic ; 

 the terms Translation-invariant, Rotation-invariant, and General-in- 

 variant are here introduced and explained ; some extensions of Dr. 

 Boole's Theorems are given, and the results finally arrived at, are 

 classified and tabulated. In the eleventh section, the lengths of the axes 

 and the area of the conic given by the general equation, are obtained 

 with ease. The twelfth section contains a very satisfactory improve- 

 ment on the ordinary method of obtaining the equation of the asymp- 

 totes of a conic ; a modification of this method, as well as some 

 applications, are added. The thirteenth section gives two methods of 

 determining the well-known equation for the eccentricity, and a third 

 method, given later on, is here mentioned. The fourteenth section 

 determines the position and magnitude of the director-circle, both in 

 rectangular and oblique coordinates ; and, in the case of the equilateral 

 hyperbola, it is proved to degenerate into the centre of the curve. In 

 the fifteenth section, two methods are given for transforming the 

 general equation, when the asymptotes are taken as lines of reference • 

 the new equation thus obtained is then geometrically interpreted. Sec- 

 tions sixteen to twenty deal with Laplace's Linear Equation to a Conic • 

 the sixteenth section treats of the genesis of the equation ; the seven- 

 teenth section furnishes the meaning of the constants involved ; the 

 eighteenth section shews the intimate connection which subsists be- 



