MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 201 



88. The secondary chords through points lying on a given diameter, and having 

 them as their mean points, being inclined at varying angles with that diameter, to 

 determine that one which makes a given angle say 6 with the diameter 



tan 6 = y"(ax"x l eyj sin oj/fay"*^ + ex'y l y"(ax"x l -} ey^ cos a>} , 



01 being the angle between the asymptotes of the centroid : viz., it is the chord which 

 meets the diameter in the point (x^y^ such that 



x l :y 1 = e (y"(cos <a sin o>) + x" tan 6} : ax" [x" (cos <a -f sin w) y" tan 6}. 



In particular, the point at which the secondary chord is perpendicular to the diameter 

 is determined by 



x l :y l = e (x" + y" cos w) : ay"(y" + x" cos ). 



The angle between a diameter and its primary chords or ordinates is given by 

 tan 4> = (ax" 3 - ey") sin <o/[x"(ay" + e) (ax" 2 + ey") cos w), 



and those diameters which are perpendicular* to their chords are the tangents to the 

 centroid at the points determined by 



x (ay + e) = (ax 9 + ey) cos o>, 

 i.e., the intersections of the conic 



a (x 2 cos a> xy) -f e ( x + y cos a) = 

 with the centroid ; or, multiplying by 6, those of the parabolas 



abx* cos at + be ( x -f- y cos o>) e 2 = 0, 

 aby 2 cos ta -\- ae(x cos <a y) e* =. 0. 



The finite intersections in question are only three in number, their coordinates being 

 determined by the cubics 



a*bx? cos w abex* ae*x + & cos o = 0, 

 ab 2 y* cos o> abey z be z y -\- e 3 cos <a = 0. 



89. When the asymptotes of the centroid are imaginary, two convenient Cotesian 

 axes of coordinates are found in the diameter parallel to one asymptote of u and its 



* " Diameter antcm ad Ordinatas rectangnla si modo uliqua sit, etiani Axis dici potcst." NEWTON, 

 ' Knnmeratio Lim arum Tortii Ordinis,' 2. 



MIXXJCLXXXVIII. A. 2 D 



