138 Dr. C. Taylor. Note of a [June 19, 



VIII. " Note of a Theory of Orthoptic and Isoptic Loci." By 

 CHARLES TAYLOR, D.D., Master of St. John's College, 

 Cambridge. Communicated by J. W. L. GLAISHER, M.A., 

 F.R.S. Received June 10, 1884. 



The orthoptic locus of a curve and its isoptic loci are the loci 

 of the points of concourse of pairs of tangents drawn to it at 

 right angles, and at angles equal to given angles, respectively. 



As a step towards a general theory of such loci, of which special 

 cases only have been treated hitherto, it is shown below that the order 

 of the orthoptic locus of a curve of class n is n(n 1), and the order 

 of its isoptic loci 2n(n 1). 



The principle on which our proof depends is that lines drawn from 

 either of the circular points at infinity I and J may be regarded as 

 intersecting at any angle whatsoever,* but such as are drawn from 

 any other point at infinity, real or imaginary, can only be regarded as 

 parallel, unless one of them be the straight line at infinity, which 

 makes an indeterminate angle with any straight line. 



1 . The Orthoptic Locus of a Curve of any Class. 



To a curve of the nth class n tangents, constituting n(n 1) quasi- 

 orthogonal pairs, can be drawn from I or J. Each of these is 

 therefore a point of the order ^n(n 1) on the orthoptic locus, and 

 this locus, having in general no other points at infinity, is of the order 

 *(-!). 



If the curve touches the line IJ in one point, n 1 other tangents 

 can in general be drawn to it from any point on IJ, and each of them 

 may be regarded as orthogonal to IJ. Every point at infinity is 

 therefore of the order n 1 on the orthoptic locus, and the remainder 

 of the locus when the factor IJ"" 1 is subtracted is of the order 

 n(n 1) (n 1), that is to say (n 1)*, and contains I and J as 

 points of the order ^w(n-l) ( !), or (n l)(n 2). 



If the curve touches IJ in r points it appears in like manner that 

 the orthoptic locus contains IJ as a factor r(n 1) times, and the 

 remainder of the locus is therefore of the order (n r)(?i ; 1), and 

 contains I and J as points of the order n(n 1) r(n 1), or 



* To demonstrate the existence of the circular point*, draw a circle, and upon it 

 take an arc AB at random, and let x be either of the points in which the circle 

 imvN the line at infinity. Any two straight lines through x may be regarded as 

 making zero angles with xA and xB respectively, and therefore as including an angle 

 equal to that standing on the arc AB, which may be of any mignitude whatsoever. 

 It readily follows that all circles pass through x, aud hence that there can be only 

 two such points on the line at infinity. 



