116 UNIVEBSITY OF COLORADO STUDIES 



As a geometrical transformation of the link-work does not affect 

 the properties of closing of the above series of circles we may state, 

 in analogy with the link- work: 



If a series of circles [generalized Steinerian series^ of the 

 prescribed kind, based upon two fixed circles C, and C^ and a third 

 circle C^ [or 6\), closes and c<>ntains 7i circles, then every other 

 series of circles based in the same manner upon the same three 

 circles closes and contains 7i circles. 



By special disposition and by assuming some of the given circles 

 as points or straight lines a great variety of circular series may be 

 obtained. If the circles C,, C^ and C3 are parts of a pencil of circles, 

 then C^ belongs to the same pencil. The series of circles obtained 

 by this arrangement have been considered by Steiner.' One of the 

 most interesting cases arises if one of the circles G^ and Cj, for 

 instance C,, degenerates into point O. All circles of the series pass 

 through O, and the circle C^ coincides with O. Any inversion hav- 

 ing O as a centre transforms all circles of the series into straight lines 

 which are inscribed in a circle C/ and circumscribed about a circle 

 C,' i. e., the limited portions of these lines form a polygon which is 

 inscribed in one and circumscribed about another circle. The prop- 

 erties of closing of these polygons, which are called Poncelet's poly- 

 gons, are the same as those of the general series of circles. Poncelet's 

 constructions also result directly from a geometrical transformation 

 of the link- work in which r,=r^. 



^12. Special Cases. 



1. I shall treat of the specializations referred to above iu detail. 

 Let r, = r2 = r, then in order to reduce the integral 



dx 



J 



V {x — a){x — b){x — c){Qi — d) 



to Legendre's normal form, we have to notice that in the case of an 

 unlimited motion 2r>R+(?, or {r — e)>(E-— r),or {r — ey>{'R — ^)^ 

 But we have also E.4-^>/'+^, and r-\-e>r — e, hence ^-\-r>r-{-e> 

 r — e>l& — r. or 



(') Werke, Vol. I, pp. 19-76 



