344 WILSON AND MOORE. 



This is a very useful standard form for the expansion of a surface near a 

 given point. Then |ix8 = 5(eki +/ko)xk3, and only the ratio e:f is 

 effective in changing the plane jJ-xS. The equation therefore contains 

 three constants after h and the plane H-xS are satisfied, namely A, B, 

 and e or /, which may suffice to determine the indicatrix with its 

 center and plane already fixed. 



Using polar coordinates (p, 6) in the tangent plane, we have 



zi = hp2(h + ccos2^), Z2 = hp'fcos2d, (109") 



Z3 = hpKAcos2d + Bsvrae), Zi = 0, i > 3. 



The normal vector distance, of the surface curve in the direction 6, 

 above the tangent plane is therefore 



y[{h + f>cos2^)ki + /cos2^k2 + {A cos29 + B 2sin0)k3], (110) 

 and the normal curvature a, is 



a = (A + e cos20)ki + / cos20k2 + {A cos20 + B sin20)k3. (Ill) 



The vectors 8 = a — h and |a, which is 8 advanced 45° in d, are 



6 = (eki + /k2 + ylk3)cos2^ + 5k3 sin2^, 



H- = - (cki + .fk2 + ^k3)sin20 + 5k3cos20. (112) 



The indicatrix reduces to a line when and only when 5 = or e = 

 / = 0. The former may be regarded as the general case. It appears 

 then that 8 may describe an_y line in the normal ^3 and the range of 8 

 may be for any distance (e" + /^ + ^4^)^ along that line. If the 

 indicatrbf does not reduce to a line, and if u, v denote coordinates 

 referred to the unit orthogonal vectors la^ and k' = (eki+/k2)/ 

 (e^ -\- P)'^, we have 



u = ^cos2e + BmV29, v = (e^ + f)kos26. 



Let e = \cos(p, f = Xsin^. The plane |i'x8 determines tp but not X. 

 The equation of the indicatrix in its plane is then 



\^u^ - 2\Auv + {A^ + B^y = B^W (113) 



Any ellipse may be written as 



aw2 + 2buv + cv^= I, a > 0, c > 0, ac - b'^ > 0. 



