428 Mi^ Buiifiside, On double-sixes. 



On double-sixes. By W. Burnside, M.A., F.R.S. 

 [Received 7 February 1910.] 



The configuration of twelve lines, known as a double-six, arises 

 naturally in connection with the theory of cubic surfaces. Its 

 connection with the theory of quadric surfaces has not, so far as 

 I know, been investigated. It is the object of the present note 

 to establish the existence of a double-six from what I believe to 

 be a fresh point of view, connecting it with a pair of quadric 

 surfaces that stand in a special projective relationship to each 

 other. A word of explanation must be given of the digression in 

 I 3. The theory of quadrato-quadratic equations has been very 

 completely worked out; but the particular relation here required 

 could not be quoted in a convenient form. It has therefore been 

 established directly. 



1. Let 8, 8' be two quadrics. On each choose one of the two 

 sets of generators : and let \ be the parameter that distinguishes 

 one of the chosen set on 8, and //, the corresponding quantity for 

 8'. A generator of 8 meets 8' in two points, through each of 

 which will pass one generator of the chosen set on 8'. In other 

 words, if \, fi are parameters of two intersectiug generators on 

 8 and 8', they are connected by an equation which- is quadratic 

 in both \ and yu,. Denote this equation by 



f{\H') = (i). 



Let yu,_i, fio be the roots of this equation when X is Xq; ^o. \ the 

 roots when /j, is yu-o, X_i, \ the roots when /j, is /jb_-^; and so on. 

 Then there arises an, in general, unending series of quantities 



. . . X_oyu._3X_i/i_i Ao/U'o^i)"'i^2 (ii)» 



all of which are rationally determinable in terms of any two 

 consecutive ones. Corresponding to this series there is an, in 

 general, open polygon whose sides belong alternately to the 

 chosen generators on 8 and the chosen generators on 8'; and 

 no side of the polygon meets any other except the two which 

 immediately precede and follow it. The theory of an equation, 

 quadratic in both variables, such as (i) is well known. Starting 

 from an arbitrary pair of values that satisfy it, the series of 

 values (ii) is in general unending, but if it is periodic for some 

 chosen initial value of X (or /x), i.e. if it is of the form 



. . . X,i/l.,i,Xo/ioXi/ii . . . i^tif^n • ■ •} 



then it is periodic for every initial value. Suppose now that 

 the quadrics 8, 8' are such that the polygon is a (gauche) 



