MOORE. — INFINITESIMAL PROPERTIES OF LINES IN .S'4. 3G1 



Two lines in S^ which intersect correspond to circles which lie on 

 the same sphere. 



A pencil of lines in /S!, corresponds to a pencil of circles on a sphere. 



All the lines in S^ which pass through a given point correspond to 

 all the circles on a given sphere. 



The lines in S^ which lie in a plane correspond to the circles which 

 pass through two lixed points. 



Developable surfaces in tS^ correspond to annular surfaces and ruled 

 surfaces to circled ,surfaces. 



We are now able to interpret the preceding theorems in S^ in circles. 



15. Five-fold infinite systems of circles (7s. In each circle C 

 of C/5 t/iere are co* linear systems A^ tangent to it. Each tangent A5 

 cuts Uz in a circled surface which has C for multiple generator of 

 order 16. 



The circles of U^ can he grouped into cc* annular surfaces such that 

 the osculating pencil of circles (the pencil determined by C and the two 

 points in which it cuts the curve corresponding to the cuspidal edge 

 of a developable) ^ has three circles infinitely near in common with U^. 

 This can be done in sixteen different ways. 



Through each circle C of U^ pass sixteen annular surfaces having the 

 pencil of spheres determined by C for tangent pencil such that the oscu- 

 lating congruence determined by three infinitely near circles of the annular 

 surface determine the sixteen osculating pencils of which C is a part. 



Through each circle C of U^ pass cc'^ circled surfaces such that there 

 are linear series A^ tangent to U^ in two consecutive circles of such 

 surfaces. 



16. Four-parameter families U^. There are 00^ linear series A ^ 

 tangent to i^^ in a given circle G. 



All the linear circled surfaces^ Ai which contain C and two circles 

 of Ui infinitely near to it generate a linear series C^. 



All the circles of U^ infinitely near C to infinitesimals of second order^ 

 in a given direction (defined by two consecutive circles) lie in a linear 

 series A 2. If the direction is varied, the linear series will generate a 

 1/5 having all the circles of a circled surface for double elements. 



Through each circle G of Ui pass oo^ directio?is such that there are 

 circled surfaces tangent to each direction (the circled surfaces contain 



* See "Circles orthogonal to a given sphere," Annals of Mathematics, 

 series 2, 8, 57. 



' The term linear circled surface is used to indicate the circled surface 

 common to five-Unear circle complexes. Linear does not apply to the order 

 or class of the surface. 



