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XXIII. — An Example of the Method of Deducing a Surface from a Plane Figure. 

 By L. Cremona, LL.D. Edin., Hon. F.K.SS. Lond. and Edin., Professor of 

 Mathematics in the University of Rome. 



(Read 21st April 1884.) 



Let there be given, in a plane ir, six (fundamental) points 1, 2, 3, 4, 5, 6, of 

 which neither any three lie in a right line, nor all in a conic ; and consider the 

 six conies [1] = 23456, [2] == 13456, [3J= 12456, [4] = 12356, [5] = 12346, 

 [6] = 12345, and the fifteen right lines 12 , 13, .,16, 23, ..,56. 



There is a pencil of cubics 1 2 23456 (curves of the third order, having a 

 node at 1 and passing through the other fundamental points) ; their tangents at 

 the common node form an involution, viz., they are harmonically conjugate 

 with regard to two fixed rays. Five pairs of conjugate rays of this involution 

 are already known ; for instance, the line 12 and the conic [2] have conjugate 

 directions at the point 1, for, they make up a cubic 1 2 23456. 



Each other fundamental point is the centre of a like involution. And also 

 on each conic [1], [2], . . . , and each line 12, 13, . . . points are coupled har- 

 monically with regard to two fixed points. The involution on the conic [1] is 

 cut by the pencil of rays through 1 ; for instance, the point 2 is conjugate to 

 the second intersection of [1] with 12, &c. The involution on the line 12 is 

 cut by the pencil of conies 3456 ; for instance, the points 12 . 34, 12 . 56 are 

 conjugate, as 34 and 56 make up a conic through 3456 ; and the point 1 is con- 

 jugate to the second meeting of 12 with the conic [2] ; &c. 



The Jacobian of a linear twofold system (reseau) of cubics 123456 is a 

 sextic K = (123456) 2 having six nodes at the fundamental points. Since any 

 reseau of cubics 123456 contains 1° a cubic k= 1 2 23456 ; 2° a cubic breaking 

 up into a ray r through 1 and the conic [1] ; 3° a cubic made up by the line 12 

 and a conic c through 3456, &c. ; we see immediately that the (sextic K) 

 Jacobian of the reseau 1° has the same tangents as the cubic k at the common 

 node 1 ; 2° and 3° passes through the intersections of r with [1], and the inter- 

 sections of c with 12, &c. 



The Jacobians K form a linear threefold system of sextics (123456) 2 



XK + X'K' + X"K" + A.'"K'" = , 

 therefore we have the following theorem : 



If six points 1, 2, 3, 4, 5, 6 are given in a plane it, as said above, we may 



VOL. XXXII. PART II. 3 X 



