140 Theory of Orthoptic and hoptic Loci. [June 19, 



4fn-ics. Thus the skew pedals of an ellipse with respect to a 

 point (regarded as a curve of the first class) are of the eighth order, 

 each consisting, of coarse, of two equal curves similar to the right 

 pedal. 



By taking a pair of lines drawn at random through either circular 

 point, which may be regarded as inclined at an indeterminate angle, 

 and supposing them to coalesce, we infer that any straight line through 

 I or J may be regarded as making an indeterminate angle with 

 itself.* 



Hence the points of contact of the tangents from I and J to any 

 curve are points on its orthoptic locus, and they are doubly points on 

 its isoptic loci. 



In the case of the conies these are the only points in which such 

 loci meet the curve. If, therefore, U=E0(a;, ?/)=0 be a conic, and 

 u=0 its orthoptic locus, the bicircular quartics which are its isoptic 

 loci will be represented by 



where k is a constant which vanishes when the isoptic angle is zero, 

 in which case the locus consists of the conic and the line at infinity, 

 and is infinite when the angle is a right angle, the isoptic being then 

 the orthoptic locus. 



Reciprocally, in a curve of the nth order, if a chord subtends a 

 constant angle at a fixed point its envelope is of the class 



The various points in the theory of plane orthoptic and isoptic loci 

 propounded in this note have been verified by analytical methods in 

 an unpublished paper by Mr. J. S. Yeo, Fellow of St. John's 

 College. 



The following notes on isoptic and other loci in space are taken 

 from a valuable and suggestive series of investigations by Mr. Joseph 

 Larmor, Fellow of St. John's College. 



A solid has in general six degrees of freedom to move. The corner 

 of a cube whose three faces are constrained to touch a surface loses 

 three and retains three, and the locus of a point rigidly connected with 

 it is not a surface, but a solid bounded by a certain envelope. When 

 the cube-angle envelopes a quadric it can enjoy one of its degrees of 

 freedom without displacement of its vertex, for if a cone of the 

 second degree has one triad of orthogonal tangent planes, it has a 



* It is sometimes said that such lines are at right angles to themselves ; but tlm 

 statement, although true so far as it goes, is inadequate. The angle between lines 



parallel toy + mx = and y + m'x=Q is tan ' - ; , and when m = m'= -S1, it is 



1 + tntn 



tan" 1 -, the numerator as well as the denominator vanishing. 



