422 



HITCHCOCK. 



plane to be .Ti = 0. By (96) this gives, (on putting p = 182, x^ = 

 xi = 0), 



An = 0. 

 If we next subtract from the vector the term 



which is of the form pSbp and does not alter the axes, we shall remove 

 the remaining term in x^xz. If the vector coefficient of c^ be called f , 

 the vector may be written 



^AAnXiX2 + (.4 12 - ^21)3:3X1] + ^xz^ (168) 



The polar vector of VpFp now becomes 



Vp'MAuXiXi + (^12 - A2i)xzx,] + ^0:3^} + Vp^,An{xi'x2 + 0:1X2') 

 + Wi(^i2 - ^21) {xz'xi + XzX^) + 2Vp^xz'xz. (169) 



If the rule for a triple axis be applied to ^2, we shall now put Bi for p' 

 and &z for p, that is X\ = xz = and .Ti = 0:2 = 0. This gives f 

 parallel to the t of the rule of Art. 18. Again, ,81 is parallel to the r of 

 the same rule. Hence if /32 is a triple axis, we must have SjS^^i^ = 0. 

 AVe may therefore take 



r = OiiSi + a2|82 

 and the vector becomes 



^i[AizXiX2 + (An — A2i)xzxi + aiXz^ + 18202X3^ (170) 



which has only two distinct axes, /Si and ^2, quadruple and triple. 

 If ai is zero, 182 is an inflectional element of the cones (3). We cannot 

 have a2 = or An = if the vector is to be irreducible. 



It appears therefore that all possible quadratic vectors which are 

 irreducible, and have j8i for a quadruple axis of the sort where all cones 

 (3) have double elements at /3i, are included under (165) and (170). 

 It is so far assumed that /3i is not of higher order than four. 



28. For Case 2° we may follow a similar method. Take j8i an 

 axis, supposed at first to be at least triple. Take (83 in the tangent 

 plane to (3) at (81. If /3i is rendered a zero the vector may be written, 

 j82 being at present any vector such that (123) does not vanish, 



