142 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



9 2 U> 

 ; ^^ + y 



9x 12 

 9 2 V 



X 



X 



1 + 9 



9 2 W 

 9 x' 9 y' 



9 2 V> 



+ 



9x' 9y 



9U' 



9x> 



9V< 

 Jx 1 



7 + 



x 



X 



9 2 U< 

 9 x' 9 y' 



9 2 V 

 9 x' 9 y' 



9 2 U> 

 9y' 2 

 9 2 V> 



9U> 

 9y> 



9 V 

 9y> 



= 0. 



This determinant and its derivatives vanish for the point P', therefore 

 the locus is a quadratic three-way cone with its vertex at PL This 

 cone is intersected by the tangent plane at P' in a pair of straight lines 

 which is the required locus. If a point x, y, . . . , be taken on either of 

 these lines, we have three independent equations just sufficient to deter- 

 mine the ratios of the four differentials ; i. e., just sufficient to determine 

 the consecutive point P", so that the tangent plane at this consecutive 

 point will intersect the tangent plane at P' in the point selected. That 

 these two consecutive tangent planes have no further intersection may 

 be further shown by forming the equation of the plane that goes through 

 their common intersection and through both the points P' and P". The 

 equations of this plane are 



A" V .&U> - A" U' . A V = 0, 

 A' V" .AU"- A' U" .AV" = 0. 



These equations in general represent a definite plane so long as P' and 

 P" are not coincident. 



It would be of interest to examine the motion of the point of inter- 

 section along these lines as the point P" circles about the point P', and 

 to see whether at any time the consecutive tangent planes intersect in 

 one of these lines. 



These lines are not inflexional tangents to the surface ; lines meeting 

 the surface in three consecutive points do not generally exist in a space 

 of more than three ways. For such lines would have to satisfy both 



A U> = 0, 

 A V = 0, 



and 



A 2 U> = 0, 

 A 2 V = 0, 



