324 



WILSON AND MOORE. 



or 



a' = acos^e + 2|isin0cos0 + psiiv^, 

 P' = pcos20 - 2K.sin0cos0 + asin-0, 

 fjL' = |i.(cos20 - sin-0) - (a - p)sin9cos6>. 



(880 



Hence if we write 



h 

 a' 



P' 

 8' 



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

 = h + |i.sin2^ + 8cos20, 

 = h — li.sin20 — 6cos2e, 

 = lk;os20 — 5sin2e, 

 = 8cos20 + |i.sin20. 



-P). 



(89) 



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

 changes, the extremity of ci' describes an ellipse of which p. and 5 arc 

 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 a; 

 and ^', laid off from the center of the ellipse, describes the same ellipse, 

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

 advanced by the excentric angle it/2 from a' toward P'. (The H- that 

 goes with the orthogonal trajectories is clearly — [J- as previously 

 proved). 



The conic, which we 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 partictdar 

 point may be described in 

 a S-space; for such prop- 

 erties surfaces in more than 

 five dimensions need not be 

 Figure 1. discussed. 



