360 PROCEEDINGS OF THE AMERICAN ACADEMY- 



The second term of this expression vanishes if (2) is satisfied ; hence 

 (^, 2. f "^^ - diiidn^ =0, k= 1, 2, 3. 



The hyperplane ($, ^) will then have three points of contact in- 

 finitely near if the ten equations (1), (2), (3) are satisfied. Tliis 

 requires one relation, and hence there are co' directions du-i of I's 

 generating a cone of order nine along which a hyperplane has three 

 infinitely near points of contact with I's. Then 



In a three-parameter familif of lines through each point pass oo * 

 directions such that a C^ ivill he tangent in three lines of the system 

 infinitely near. 



IV. Application to the Geometry of the Circle. 

 14. If the equation of the sphere is written in the form 



t (.r^ + / + C-) + 2 {ax + hy -^ cz) + 2 d = 0, 



the quantities a, b, c, d, t may be taken as the coordinates of the 

 sphere. The spheres of the pencil determined by two spheres (a, h, 

 c, d, t), {a, U, c\ d', t') will have coordinates 



{a + Xa', h + Xh\ c + Xc, d + Xd', t + Xt'). 



Then, if we consider the circle defined by this pencil as representing 

 the pencil, we see that the circle in ordinary space corresponds to the 

 line in /S'j joining the points 



(«, b, c, d, t), (a', b\ c, d', t'). 



The coordinates of the circle can then be taken as the two-row de- 

 terminants of the matrix 



a b c d t 

 a' b' c' d' t' 



These coordinates are identical with those used of the line in *S!j. 



The lines in /S^ which lie in an /S's correspond to the circles which cut 

 a given sphere orthogonally.'^ 



' For the explanation of this and the following statements the reader is 

 referred to the paper of the author previously referred to. 



