616 TRANSACTIONS OF SECTION A. 



Also F (105) is (algetraically) resolvable into two large factors, say L, M, in 

 following cases :^ 



Of (i/"5 + 1), when ?/ = 3^^^ Ir,-, Ibr,", '6bri^ 



Of (j/^o' - 1), when y = Zrf-, 2\rf, IOSj;'' 



and each factor L, M is of order y-*, and is susceptible of the six 2'<= partitions 

 (<= + Xm8), (i-^-X'M-), whereX = 3, 7, 35; X' = 5, 21, 105. 



The values of the large factors L, M of F (105) are shown below — resolved 

 into their prime factors as far as found practicable — for the small base 

 J/ = 3, 5, 7, 12. 



N (105) L and M of P (105) 



(3'°=+l); L =24151 . 3369031 ; M-211 . 1051 . 3454081; 



.K,a5_iN /L =21, 226, 783, 250, 214, 361* 

 ^'^ ' 1m = 307, 468, 970, 805, 907, 721* 



/•7105 . IN /L =211 . 338, 640, 865, 331, 157, 691t 

 ^ "^ ' 1 M = 460, 798, 894, 542, 150, 330, 401; t 



np'os , u f L =46, 199, 145, 931, 519, 207, 045, 842, ISlf 

 ^'■'' '^ ^^ 1 M = 1471 . 85, 890, 295, 500, 677, 161, 086, 861t 



The composition of the large factors in the L, M above is unknown ; but those 

 marked * contain no factors < 100,000, and those marked f contain no factors < 10,000. 



The complete factorisation (into prime factors) of the seven algebraic aub» 

 factors of the above X (105), viz., of 



(y T 1), (/ T 1), (y> T 1), {y' T 1), {y^'" T 1), (i/« T 1), (f- T 1) 



is known for each of the above bases [y = 3, 5, 7, 12], except that of (12'* + 1), 



G. An Elementary Discussion of SchlcifiVs Double-six, 

 By Professor A. C. Dixon, Sc.D., F.R.S. 



The object of this note was to discuss by elementary methods the system of 

 twelve straight lines in space known as a double-six. The lines lie on a cubic 

 surface,' but no reference is made to this fact. 



Let a be a given straight line and 2, 3, 4, 5, 6 five other lines meeting it in 

 c(,, a^, a^, a.^, a,, respectively. Each of the five tetrads 3456, 2456, 2356, 2346, 

 2345 is met by a second line ; let these be b, c, d, e, /, so that b meets 3, 4, 5, 6 in 

 ^3, b^, 6j, 6g respectively, and so on. The theorems to be proved are these : — 



1. The five lines b, c, d, e, f all meet another line, to be called 1. 



2. The pairs of planes connecting b with flj, c^ ; a^, d.^; 0^,6.^; «g,/j are in 

 involution: there are sixty relations of this type. This theorem includ^ea (2')? 

 that (c, d., e.,f.,) = (Kg a^a^ a^). 



S. If the intersection of the planes o2 and 61 is called (12), and so on, the 

 lines (12) (34) (56) lie in a plane : of such planes there are fifteen. 



The theorem (2) is new to me; the others are known in the theory of the cubic 

 surface. 



' Salmon, Geometry of Three Dimeiisiom, 4th edit., p. 500. Mr. Richmond has 

 lately drawnattention to the double-six, Camh. Phil. Proc., vol. xiv. p. 475. For an 

 elementary discussion of the generators of a hyperboloid, without reference to 

 the surface, see a paper in this year's Prpceeduigs of the Edinburgh Mathematical 

 Society. 



