2 Colorado College Studies. 



given lines in a plane, a point Pin, and for every odd number 

 a circle Con+i. with the property that always the point P^„ is 

 common to all the 2n circles Czn-i, and always the circle 

 Cin+i passes through all the 2w + l points P^n. 



Clifford's method is indicated by Salmon,* but as the lat- 

 ter author does not enter upon the above-mentioned extension 

 to more lines than five, I take the liberty of summarizing the 

 proof of Clifford, though his original paper, which no one of 

 geometrical tastes should omit to read, is fairly accessible. 



An n-fold parabola is defined as a curve of class 7i-f 1, 

 touching the line infinity n times. Such a curve is rational 

 and of order 2//, and is determined by 2n-f-2 finite tangents. 

 It has only one focus; for from the circular point /there can 

 be drawn only one tangent beside the line infinity, and this 

 tangent meets its conjugate from/ in the single real point i''. 

 If the number of given finite tangents is only 2?i-|-l, a single 

 infinity of curves can be drawn, and the focus of each is the 

 intersection of a ray from I by the projectively correspond- 

 ing ray from ./; hence the locus of i'^ is a conic through / 

 and/y that is, it is a circle. Among the curves of the above 

 pencilf there are 2?j + l cases of disintegration, viz , an M-fold 

 parabola may consist of the point in which one of the given 

 lines meets the line infinity, together with the (?j — l)-fold 

 parabola touching the other 2n lines; and its focus is the 

 focus of the latter parabola. Hence the 2n + l foci of these 

 special curves lie on the one circle determined by the 27i-|-l 

 given lines; while, when 2?i-t-2 lines are given, we may from any 

 2?i-4-l of them determine one circle as above, and all these 

 circles will pass through one point, viz., the focus of the 

 ?t-fold parabola determined by all the lines. 



Clifford does not consider, (as do Kantor and P. SerretJ) 

 the condition under which Miquel's circle breaks up into a 

 right line plus the line infinity, but a problem essentially 

 similar is treated by Salmon .§ The statement for the general 



* Higher Plane Curves, p. 128. 



t Curves forming a pe«ci7 are usually understood to have in common a number 

 of points, one less than suffices to determine the curve ; here, however, and through- 

 out this paper, substitute for common points in this definition, common tangents. 



tComptes Rendus, 189. 



§ Higher Plane Curves, § 145, p. 127, 



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