Cliffords L^roof of MiyuEi/s Theorem. -i 



case may be shortly made as follows: The circle which is 

 the locus of the foci of w-fold parabolas will break up in this 

 way if the pencil of such paraljolas contains one that has 

 the circular points as two of its contacts with the line in- 

 finity. For that parabola can have no finite focus. Thus the 

 Mitjuel circle of five ^iven lines is replaced by a rij^ht line if 

 the five lines are all tani^jents to a curve of class three, order 

 four, touching the line infinity at the circular points. This 

 curve is the hypocycloid of three cusps, by some called the 

 deltoid. 



A number of theorems may readily be reached from the 

 suggestion afforded by Clifford's demonstration. 



Thus we might discuss pencils of curves having two real 

 foci, one of which is fixed, and seek the locus of the remain- 

 ing one. A series of theorems would thus be derived where 

 the locus found would in general be a circle; though in the 

 first member of the series, namely, the case of the conic 

 touching two lines and having a fixed focus, the locus is recti- 

 linear.* Passing by such partial, though perfectly valid, 

 applications of Clifford's method, I wish to notice a case 

 which I regard as interesting, in which a true result is sug- 

 gested rather than proved by an argument framed in imita- 

 tion of Clifford's. 



Let us consider a curve of third class and fourth order, 

 having a point of contact with the line infinity, and also 

 meeting the same in the two circular points. It is rational, 

 and has no inflection, one double tangent, three cusps and 

 no other double point. No tangent which shall touch the 

 curve elsewhere can be drawn through / or J, save the line 

 IJ itself; there is therefore no simple focus analogous to 

 that of the conic parabola, such as any 7z-fold parabola 

 possesses, but there is one and only one focus, the meeting 

 point of tangents at / and J, and thus of the type exempli- 

 fied by the center of a circle. The number of conditions 

 imposed by the definition of the curve is four, — two given 

 points, one given line as tangent, and an unspecified double 

 tangent. Five more are required to determine a curve, or 



* StilmOD, Conies, v. 320. Ex. 3. 



