38 PROCEEDINGS OF THE AMERICAN ACADE3IY. 



erator g'. As there is a finite number of pairs of values of x and y, we 

 may arrange them in order of magnitude, calling g the greatest, i. e. we 

 shall say that g is the greatest number of points of C*"* on any generator ; 

 correspondingly, / is the smallest number of points of C'"' on any gen- 

 erator, since g + g' = k. Now a plane through D' cuts out two genera- 

 tors, and as we revolve the plane about i)' the number of points of C'"* on 

 the two generators in the plane is constant and equal to a — 8', where 

 h' is the number of points of C""' on U . If then we pass a plane through 

 jy and a generator g' , having the least number of points of 6'"" on it, 

 the other generator in this plane must have the greatest number g of 

 poiuts of C'"> on it ; therefore g -Y g — a — 8', and since g ->r g' — a—h, 

 we have 8 = 8', i. e. 6'"" meets each of the double directors, D and D', 

 in the same number of points, 8. If now a plane be passed through a 

 generator g and the director D it will cut out a generator g , and if 

 through this generator g and U we pass a plane it will cut out another 

 generator g, so that there are at least two generators g. Through these 

 two generators g, the directors D and D (which four lines form a 

 gauch-quadrilateral), and an arbitrary point we can pass a quadric ; 

 D, D', and the two generators counting for 2 + 2 + 1 + 1 = 6 lines in 

 the intersection of the quadric and scroll. Each generator meets the 

 quadric in two points, one on D and one on U , and if we take the arbi- 

 trary point on any generator, this generator will lie on the quadric, and 

 tlie remaining intersection of the quadric and scroll will be another gen- 

 erator. Thus by varying this arbitrary point all the generators of the 

 scroll, two at a time, will be successively cut out by a variable quadric 

 tliat always contains D, Z/, and the two chosen generators g. This 

 quadric always meets C'"' in 2 a points, of which 2 8 lie on Z) and IJ 

 and 2g lie on the two chosen generators g, so that the remaining 

 2a — 2h — 2g points lie on the other two generators cut out by the 

 (juadric ; but a — h ^ g -\- g' and therefore 2 a — 2 8 — 2 ^r = 2 ^'. 

 Now there is no generator that has fewer than g' points of C'"' on it and 

 the number of points of C'"' on the two generators together is 2 ^' ; 

 therefore each must have g' points of C"' on it. Therefore every gen- 

 erator of the scroll, except the two chosen generators g, has g' points of 

 C<") on it. In like manner, if we choose two generators g', through which 

 the variable quadric is always to pass, we have the sum of the number 

 of points of CW on the other two generators cut out by the quadric 

 always equal to 2 a — 2 h — 2 g' =z 2 g, and, since no generator has 

 more than g points of C(') on it, every generator on the scroll, except 

 the two chosen ones, has g points of C'(") on it. Therefore g ~ g' and 



