( 537 ) 



points in the six basepoints and twofold points in the ten meeting 



points of the triplets of lines. 



The remark that the curve c 12 belonging to the six basepoints of 

 tig. 1 has the line c' lt as axis of symmetry and transforms itself 

 into itself when rotating 72° around A\, enables us to deduce in a 

 simple way its equation with respect to a rectangular system of coor- 

 dinates with A' t as origin and c' 18 as .i'-axis. The forms which pass 

 into themselves by the indicated rotation are 



q* = x' -f y\ P=zx' — 10#y + 5.iu/ 4 , Q = hx A y — \0xhf -f f. 



If we pay attention to the axis of symmetry and to the identity 

 P* -J- Q s = j> 10 the indicated equation can be written in the form 

 q* + ao° -f- b Q » -f co 10 + d 9 " + P(e + /q' + yQ* + ho°) + P 2 (/ + /-o s ) = 0, 

 so that we have to determine only the ten coefficients a, h, . . , k. If 

 now the common distance of the points A\, A'„ . . , A', to A\ is 

 unity, then 



« ' 



3 -A 2 / 3 + e\ 



where e stands for |/5, represents the twelve points of intersection 

 of the curve with the .r-axis. By performing the multiplication this 

 passes into 



sb a (x* + 2x' — 7x e — 6x* + 20« 4 — Qa} - lx* 4-2* + 1) = 0. 

 From this follows 



a = — 7, & = 20, c-\-iz=z— 7, d + k == 1, 

 e = 2, ƒ= — 6, g=-Q, / t = 2. 



So the equation 

 Q * - 7 9 '4-20 p 8 - 7 9 10 4-o 12 4-2P(l - 3o 2 -3 ( V4V)- Q 2 0'W) = 

 is determined, with the exception of the coefficients i and k still 

 unknown. Now the parallel displacement of the system of coordinates 

 to A\ as origin furnishes a new equation, of which the form 

 (4_i_/;) iV »4_2(12 — 4«'-5%7/ 2 4-.r 4 -f(54-28 i -45/;)<y a 4-(54-4/+3% < 

 represents, after multiplication by 25, the terms of a lower order 

 than five. The new origin being a fourfold point of r 12 and the 

 terms with f and xy* having thus to vanish, we find 



ï = 8 , k z=z — 4, 



on account of which the indicated form passes into 



(«" 4- 5// 2 ) 2 . 



The correctness of this result is evident from the following. Just 



as the two tangents in the old origin counting two times are represented 



'\ v '' 2 H - 'f = 0, and therefore coincide with the tangents out of A\ 



to the conic through the other basepoints, so x* -\- of = represents 



