1887.] A. Makhopadhyay — Memoir on Plane Analytic Geometry. 297 



the same in all the equations, the system is concentric ; if c alone is 

 the same in all the equations, the circles are such as can be intersected or- 

 thogonally by a circle of radius V c, described round the origin as 

 centre ; and this shews at once that as a system of co-axal circles can 

 be orthogonally intersected, their equations must necessarily be of the 

 form 



x^-^y^-2Jca;=z±B% 



where S is constant, but k variable. 



The geometric meaning of c also furnishes the length of the tang- 

 ent drawn from any point to a circle, for, the equation of the circle 

 being 



x'-{-y^+2gx + 2fy-{-G = 0, 

 and the point from which tangents are drawn being (x', y'), remove the 

 origin to this point ; then, the new absolute term is clearly the point- 

 function of the circle with respect to the point (£c\ y'), and this, 

 therefore, is the length of the tangent sought. It follows as a conse- 

 quence of this, that the geometric meaning of the equation of the 

 circle is that, if the length of the tangent drawn from any point to a 

 circle vanishes, that point must be on the curve itself. 



§. 7. Chords and Tangents of Circles and Conies.— The fol- 

 lowing equation of the chord joining the two points (oj', ?/'), (x", y'\) on 

 the circle 



^«4-7/ = r' (26) 



is due to Professor Burnside, (Salmon's Goriics, §. 85, Ed. 1879, p. 80), 



(^-:.0(^-^'7 + (2/-/)(2/-2/")=^Hr-^' (27) 



It is easily verified that this is actually the equation of the chord ; the 

 following geometrical interpretation, however, shews the genesis of the 

 equation. 



On the line joining the points (x', y'), (x", y") as diameter, describe 

 a circle ; any point (x, y) on this circumference is such that the lines 

 joining («, y), (x', y'), and {x, y), (x"^ y"), include a right angle ; this 

 condition, expressed analytically, gives for the equation of the circle 



(^-^')(^-^")-i-(2/-2/OOy-2/'0 = o (28) 



The chord in question may now be regarded as the common chord of 

 the two circles represented by (26) and (28) ; and then, from the ele- 

 mentary principle that S + A;S' = represents any locus through the 

 common points of S = 0, S' = 0, we at once write down Burnside's 

 equation (27), the proper value of k being easily seen to be given by 



1+7j = 0. 

 The generalisation to the conic given by the general equation 



ax^-\-2hxy'^by^-^2yx-\-2fy-\-G=:0 (29) 



38 



