Clifford's Proof of Miquel's Theorem. 5 



This theorem is true, as I have elsewhere shown analyti- 

 cally;* and is, moreover, the first term of a series of theorems 

 which would be obtained by replac-ing the curve of third order 

 in the foregoing argument by curves of higher order precisely 

 according to the analogy of Clifford's demonstration. The 

 argument here given is, however, incomplete as a proof, until 

 it is sIhjwu that the pencils of /-tangents and ./-tang(Mits 

 break up, as indicated, into separate simple pencils whose 

 rays have a one-to-one projective relation. 



To do this, by purely geometric reasoning would probably 

 not be easy; at least it may be expected that the demonstra- 

 tion would acquire a length and cumbersomeness which 

 would render it entirely unlike the elegantly simple argu- 

 ment in which the analogous theorem was established by 

 Clifford. 



♦"Sundry Metrical Tlieorems concernins ii Linos in a I'lane;" a paper read 

 before the Am. Matli. Soc. April 28, 1900, and subsequently published in the JV«7is- 

 artiu7is of that Society, Vol. 1, No. 3. 



May 1, 1900. 



