( 556 ) 



an arbitrary point Q v/a and we bring through thai point surfaces 

 /•;. F t and F u of the pencils F . F t and /-', . Through each of 

 the a — 1 points of intersection of those surfaces not situated on the 

 base-curves of those surfaces we bring a surface F r . These a— 1 

 surfaces F r intersect the right line / together in {a — \)r points Q r , 

 which we make to correspond to the point Q stu . The coincidences 

 of this correspondence are: 1 st the points Q rstu determining four 

 surfaces which intersect one another once more in a point not lying 

 on the base-curves, thus the n points of intersection with the surface 

 /,, 2 nd the points of intersection with the surface R sta belonging to 

 the pencils (F,), [F t ) and (F„), the locus of the points S determining 

 three surfaces whose tangential planes in S pass through one line. 

 To find the number of coincidences we have to determine the 

 number of points Q s , u corresponding to an arbitrary point Q, of I 

 To this end we take on / a point Q tu arbitrarily and bring through 

 it an F t and an F u . Through each of the b points of intersection 

 of these surfaces with the surface F r through Q, (not lying on the 

 base-curves) we bring an F s , which b surfaces I' intersect together 

 the line I in bs points Q s which we make to correspond to Q lu . 



To find the number of points Q lu corres] ling to an arbitrary 



point Q 8 of / we take Q u arbitrarily on I, we bring through Q s an 

 F s and through Q u an F u and through each of the c point- of inter- 

 section of those surfaces with F r an F t , which furnish c surfaces 

 Ft cutting / in ct points Q t ; reversely to Q t belong du points Q u , 

 so that we find between the points Q u and Q a (c/, ^-correspond- 

 ence, of which the ct-\-du coincidences give the points Q tu belong- 

 ing to the point Q s . So between the points Q lu and Q s exists a 

 (/ WjC f + ^-correspondence, of which the coincidences consist of 

 the r points of intersection of / with the surface F r through Q r and 

 of the points Q stu corresponding to Q r ; the number of these thus 

 amounts to bs -j- ct -{- du — r. 



So between the points Q stu and Q, there is an (ar — r,ft*+c^+^« — r )- 

 correspondence with ar+bs+ct+du— 2r coincidences. To find out 

 of this the number of points Q rstu we musl first determine the order 

 of the surface R sta ■ 



This surface may be regarded as the surface of contact of the 

 surfaces of the pencil (F s ) with the movable curves of intersections 

 C lu of the surfaces of the pencils [F,) and (F u ) 1 ). So the question is: 



l ) We shall call this surface the surface of contact of the three pencils meaning 

 by this that in a point of this "surface of contact 1 ' the surfaces of the pencils, 

 though not touching one another, admit of a common tangent. 



