Februaby 4, 1916] 



SCIENCE 



155 



(2, 2) correspondence between the para- 

 meters. Hence there should be 2 -[- 2 

 closures, whatever the degree of remoteness 

 demanded between first and last points. 

 But exactly four, improper indeed, are 

 supplied by the participation of the four 

 common points or the four common tan- 

 gents. The relevant algebraical equation 

 for the parameters will always have four 

 roots relating to improper or degenerate 

 polygons. If it has any more than these 

 four, it admits all parameter-values as 

 roots. Hence one actual proper polygon 

 of m sides proves the existence of countless 

 others. This brief but conclusive reason- 

 ing gives the problem its true setting in 

 advance, but leaves for other methods the 

 question of the existence of the all-impor- 

 tant first proper polygon. 



Grino Loria, in his memorable work, II 

 passato id il presente clelle principali teorie 

 geometriche, makes mention of these papers 

 of Hurwitz at the climax of his paragraphs 

 on theorems of closure; and says of the 

 earlier essay, that in it "we do not know 

 whether to wonder more at the immensity 

 of the view, or at the perfection of its 

 beauty ; and so with this we bring to an end 

 this digression, for which we should seek 

 in vain a close more worthy. ' ' I have pre- 

 ferred however to summarize it earlier, in 

 order to make clear with the greater brevity 

 certain other applications that depend 

 upon the same principle. 



It is hardly needful to remind you that 

 the (2, 2) correspondence leads inevitably 

 to elliptic functions, as Buler long ago 

 pointed out. If we picture the situation 

 by means of a Riemann surface, it must 

 have two leaves and four branch-points; 

 and is therefore of deficiency one, whence 

 all functions belonging to the surface are 

 doubly periodic. Of course in the fore- 

 going survey we have been thinking mainly 

 of real points and lines and loci, and so 



have neglected the second period — the first 

 being real. The use of elliptic functions 

 enables us to understand the situation in- 

 volving imaginary arguments, as when the 

 point locus is completely enclosed by the 

 line-locus, so that a real polygon is obvi- 

 ously impossible, and yet the invariant con- 

 ditions may be satisfied. The one essential 

 premise is in every case, that the things 

 under consideration are algebraically con- 

 nected, two values of either to every one 

 value of the other. 



First let me recall the chain of circles 

 devised by Steiner, most recently so inter- 

 estingly treated by Professor Emeh by the 

 aid of his mechanical linkages. Let two 

 circles enclose a ring-shaped area in the 

 plane, and draw any one circle in that ring 

 tangent to the first two. Let a second be 

 drawn touching both the directors and the 

 last mentioned circle, then a third touch- 

 ing in the same way the second, and so on. 

 If a last circle ever appears in the series, 

 say the nth in order, touching the first one, 

 call the chain closed. This chain is now like 

 the Poncelet polygon in the essential fea- 

 ture, in that every member (circle) is pre- 

 ceded by one and followed by one definite 

 member of the series : the correspondence is 

 certainly algebraic and (2, 2). Therefore 

 the chain wiU close with n circles, no matter 

 what one be selected for the first. Both 

 Hurwitz and Emch have stated weaker con- 

 ditions that lead to the same conclusion; 

 but it would seem, if the analogy of the 

 polygon porism is valid, that many other 

 variations of conditions ought yet to be 

 attempted. 



There are Steiner 's polygons on a plane 

 cubic, wdth alternate sides passing through 

 one of two selected fixed points on the 

 curve. This curve, with points represented 

 in elliptic functions of a parameter, might 

 seem out of place among conies and other 

 rational curves, but the next example will 



