Ball — On the Character of the Linear Transformation. 535 



Let, as before, a, j^, y, S be the four roots of the equation in p ; then 

 "we must have some relation of the type 



ay - ^8 = 0. 



But we have also for the displacement of a point on ay, whose co- 

 ordinates are 



tti + Oci, tti + 6c2, ttz + Oc^, a^ + dci, 

 the points 



a^ a^ A°^ /i" 



tti + 6- Ci, Uo + a- C2, a^ + v- Cz, a^-^d- C4,; 



y y y y 



and since the comers of the tetrahedron lie on the absolute, the distance 



a, 

 moved is proportional to log -, whence we deduce for the pitch the- 



extremely simple equation 



log a — log y 



log /3 - log S* 



In the case of a vector where the pitch is ± 1, we have 



log g - log y ^ ^ 

 logyS-logS ~ ' 



and, of course, ay = yS8. 



"We hence see that either 



a = fS and y = S, 



or a = 8 , , ^ = y. 



This leads to another interesting theorem, which may be thus 

 stated : — 



When the movement is a "vector," then the necessary and the 

 sufficient condition is, that the equation for p shall be a perfect 

 square. 



I take the opportunity of adding a theorem, which expresses the 

 distribution of pitch upon the surface in elliptic space, which corre- 

 sponds to the cylindroid in ordinary space. 



The equation of the surface in its simplest type is 



Xi Xo {z^ + x^) = Xi Xi {xi^ + X.}) y 



