( 348 ) 



SX.S, V s ', -I ;.._, \, -| X s S 9 = 0, | 



.... (2) 

 ^ a, S A X x S 4 i ;., S s -f K S e = O ) 



and we can easily deduce from them a theory pretty well detailed 

 of the congruence of these quadrisecani circles of y 6 . Tims we see 

 that this congruence is of order one and of class three; thai those 

 circles which pass through a fixed point of y 6 generate a eyelid 

 having this point as a node: that those circles of which the plane 

 passes through a fixed point generate a surface of order seven; we 

 find the surface which is generated by those circles resting on a 

 right line or on a curve, etc., etc. The mosl important result is thai 

 the planes of these circles envelop a cubic scroll Sc t . 



Every sphere containing a quadrisecani circle of y a has an equation 

 of the form 



2X l {S l -kS 4 ) = (3) 



and it contains the two points for which we have S t = kS 4 , S, = kS 6 , 

 S i = kS 6 . When h varies these two points describe the fundamental 

 involution discussed by Prof. .1 \\ dk Vries ; we simply shall call 

 couples, the couples of points of this involution. The chords of 

 couples are the rectilinear generatrices <>f the surface Sc 3 . 



Among the surfaces «, N, ,s', circumscribed abonl y„, we must 

 distinguish the two following 



'VV*. =0, a 4 S,S 4 =0 ..... . (4) 



They are circular cubics and they determine a pencil the base of 

 which completes itself by means of the circle at infinity and of the 

 line 



«, ", N . ' s , =0; (5) 



by subtraction of the columns this matrix is reduced to n i a 4 s. s 4 

 and represents a right line q, quadrisecani of y„, double line of the 

 surface Sc 3 and meeting all the chords of the couples. 



2. Let us write the equations of the sphere S t in the more com- 

 plete form 



% ai(x B ' + y* -f ««) -f h t ,r + ei y + d t z -f ƒ•= 0, 



let us put the matrix 



.1/ ai hi a difi [i -. 1,2,3,4,5, 6) 



and let ns call .)/, Ihe determinant deduced from this matrix by the 



omission of the row of rank /' and affected by the sign -j- or 



according to i being even or odd ; we shall then have evidently 



2 Miai= 2 Miti=SM i e i =SMid i z=SMi/i= 0, > 



. . (6) 

 2 M { Si= 0. (/' — 1 , 2, 3, 4, 5, 6) I 



