28& REV. T. p. KIRKMAN ON LINEAR CONSTRUCTIONS. 



through one of the point's positions; it is also a surface of 

 the second order, since no line can meet it once only, or 

 more than twice. That this locus passes through all the 

 nine given points, is proved by the same a' gument when 

 the paradigm is treated as a purely geometrical datum, as 

 that which has been adduced when we considered it as an 

 analytical datum. The expression, namely, 



is zero, whenever the tenth point coincides with any of 

 the given nine; for the solids that form it destroy each other 

 in pairs. 



If Xq 3/0 2-0 be any tenth point on the surface, the term A 

 can be reduced by the drawing of fourteen pairs of parallels 

 to the form A n: X^-i^^ x ; and every term in the paradigm 

 can be by the same labour reduced to the same form ; the 

 addition of the lines x x^ x-j, x^j &c., thus found on the 

 axis of X must give the result. 



r^rf^' {x + a:, +. a:, + X3 + ... ) = 0, 



which is the condition, both necessary and sufficient, in 

 order that the ten points, 1 2 3... 9 0, should lie in a surface 

 of the second order. The addition of these lines x j^„ &c., 

 is effected by drawing certain lines, in sets of three each, 

 the last of which lines will pass through the origin. 



If, then, any eleven points in space be taken, the condition 

 that any ten shall lie on a surface of the second order is, iliai 

 a line, found by the ruler only, shall pass through the eleventh. 



This is a })urely geometrical solution, by the aid of the 

 ruler only, of the Brussels Academy's prize question. The 

 found line is one of an infinite number forming a pencil 

 through the eleventh point — the particular line depending 

 on the axes drawn through that point. 



The paradigm of the general surface of the n^^ order can 

 always be transformed by drawing of given lines with the 

 ruler — the axis of x being drawn through the point (xq, 0, 0), 



