344 WILSON AND MOORE. 



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

 given yoint. Then H.x8 = J5(eki +/k.2)xk3, and only the ratio e:f is 

 effective in changing the plane |Xx6. The equation therefore contains 

 three constants after h and the plane \i-x^ are satisfied, namelj^ A, B, 

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

 center and plane already fixed. 



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



si = hP2{h + ccos2d), 22 = y^fcos2d, (109") 



Z3 = \p^{Acos2d + 5sin20), Zi = 0, i > 3. 



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

 above the tangent plane is therefore 



Ip'lih + pcos20)ki + /cos20k2 + {A cos20 + B 2sin9)k3], (110) 

 and the normal curvature a, is 



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



The vectors 8 = a — h and jx, which is 8 advanced 45° in 6, are 



8 = (fki + ./"k. + .4k3)cos20 + ^ks sin20, 



|Ji = - (eki + .fk2 + /Ik3)sin20 + 5k3cos20. (112) 



The indicatrix reduces to a line when and only when B = ox e = 

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

 then that 8 may describe any line in the normal S3 and the range of 8 

 ma}^ be for any distance (e^ + .P + -4")^ along that line. If the 

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

 referred to the unit orthogonal vectors ks and k' = (cki + /k2)/ 

 (e'-f-/*)i, we have 



u = ^cos20 + Bsin2d, v= (e^ + f)kos2d. 



Let e = \co&(p, f = Xsin^?. The plane ^'•^8 determines (p but not X. 

 The equation of the indicatrix in its plane is then 



XV - 2\Auv + (.42 + B^y = B^W (113) 



Any ellipse may be written as 



au^ + 2buv + cv"^ =1, a > 0, c > 0, ac - b^ > 0. 



