104 



Mr Brill, On the Generalization of [Feb. 24, 



5. Suppose that 



Oi, y±, z,), (x,, y 2 , z 2 ), (x 3 , y s> z,), (w 4 , y 4 , z A ) 



denote the coordinates of four points, so situated that the origin of 

 coordinates lies within the tetrahedron formed by them. We 

 will write 



2X a = 



2Z,= 



We will also write Y's with the same subscripts to denote 

 what these expressions become when for the y's are substituted 

 the corresponding z's, and for the z's the corresponding os's. Also 

 we will use Z's with the same subscripts for the expressions 

 obtained when for the y's are substituted the corresponding x's, 

 and for the z's the corresponding y's. 



In the above it is to be understood, as before, that our axes of 

 coordinates constitute a right-handed screw system. Also the 

 corners of the tetrahedron are understood to be so placed that the 

 cyclical order of the subscripts in the expression for any X is sup- 

 posed to be that given by a right-handed rotation about the 

 outward drawn normal to the corresponding face of the tetra- 

 hedron. 



We have 



z 1 +x a +x,+z 4 =o J 



Y,+ F 2 +F 3 +F 4 = 0, 

 Z,+ Z. 2 + Z 3 i Z, = 0. 



Therefore, if we write 



m 1 = X 1 +pY 1 + qZ 1 , 

 m 2 = X 2 + p F 2 + qZ. 2 , 

 m 3 = X s +pY s + qZ s , 

 m i = X i +pY i + qZ i , 

 we have 



nth + w 2 + m 3 + m 4 = (20). 



