1884.] Theory of Orthoptic and Jsoptic Loci. 139 



Notice in verification the case of the conies (de la Hire, 1685), 

 and likewise that of the cardioid, whose orthoptic locus consists of a 

 circle and a bicircular quartic, which together make up a tricircnlar 

 sextic. When the curve resolves itself into n point-factors the 

 orthoptic locus evidently consists of the \n(ii 1) circles described 

 on the lines joining the points two and two as diameters. 



2. Pedals of a Pair of Curves. 



The locus of the vertex of a right angle whose arms envelope two 

 curves of class m and class n respectively may be called the pedal of 

 the two curves, or of the one with respect to the other, and the 

 corresponding locus generated by the vertex of any other constant 

 angle may be called a skew pedal of the two curves, or of the one 

 with respect to the other. The former locus becomes a pedal 

 commonly so called when one of the curves degenerates into a 

 point. 



From the reasoning used above it is evident that the pedal of two 

 such curves is an mn-circular 2mn-ic. 



This may also be deduced from the formula for the orthoptic locus 

 as follows: The two curves make up a curve of class m+n, whose 

 orthoptic locus is the aggregate of the pedal and the orthoptic loci 

 of the two curves. The pedal is therefore of the order 



(m-\-n)(m-\-n 1) m(m 1) n(n 1), 



that is to say, it is of the order 2mn ) and it contains I and J as points 

 of the order mn. 



3. Isoptic Loci and their Reciprocals. 



a. Any two tangents to a curve from I or J may be regarded as 

 intersecting at angles and TT or these reversed, and their point of 

 concourse thus belongs doubly to the corresponding isoptic locus. 

 The order of such loci is therefore double of that of the orthoptic 

 locus, and they pass twice as often through I and J. 



For example 



(1.) The ellipse, to which one pair of tangents only can be drawn 

 from I or J, may be regarded as subtending any angle or its supple- 

 ment at those points. These are therefore double points on the 

 corresponding isoptic locus, which is accordingly a bicircular quartic. 



(2.) The parabola may be regarded as subtending any angle or its 

 supplement at every point on the line at infinity. Its isoptic loci 

 therefore contain the factor IJ 2 , and the remainders, when this factor 

 is rejected, are hyperbolas (or ellipses). 



(3.) It may be deduced from the formula for isoptic loci, or proved 

 directly by the method used above, that the skew pedals of a pair 

 of curves of class m and class n respectively are 2m?i-circular, 



