324 



WILSON AND MOOKE. 



or 



a' = acos^e + 2|Asin0cos9 + psin-0, 

 P' = pcos^^ - 2|isin0cos0 + asin-0, 

 |a' = \i.{coii-d - sin^e) - (a - p)sin^cos0. 



(880 



Hence if we write 



h = i(a + P) and 5 = i(a 

 a' = h + \i.sm2e + 8cos20, 

 P' = h - |Xsin20 - 8cos2e, 

 |x' = jJLcos2^ - 8sin2e, 

 8' = 8cos20 + |isin2^. 



-P). 



(89) 



35. The indicatrix. From equations (89) we infer that: As 6 

 changes, the extremity of Ol' describes an eUijJse of which |X and 8 are 

 conjugate radii and of which the center is given by h; the extremity of P 

 describes the same ellipse at the opposite end of the diameter from d; 

 and \>-', laid off from the center of the ellipse, describes the same ellipse, 

 each position of \^' being conjugate to the line joining a' and P' and 

 advanced by the excentric angle 7r/2 from a' toward P'. (The |JL that 

 goes with the orthogonal trajectories is clearly — |X as previously 

 proved). 



The conic, which toe thus get, lying in the normal space, may be called 

 the Indicatrix. In four dimensions the whole figure including a and P 



lies in a plane, namely the 

 normal plane; in higher 

 dimensions the figure will 

 not generally lie in a 

 plane, the ellipse with the 

 lines a' and P' forming a 

 conical surface lying in a 

 normal three space. No 

 matter how many dimen- 

 sions a surface may lie in, 

 the properties of normal 

 curvature at any particidar 

 point may be described in 

 a ?t-space; for such prop- 

 erties surfaces in more than 

 five dimensions need not be 

 Figure 1. discussed. 



