396 



HITCHCOCK. 



In a similar manner \vc may obtain any other A with double subscript 

 in terms of k. Thus 



.432 (456) - SK^vC\,eiCS, 2) - Sk^z- T,,,,{n, 2), (76) 



where, as before, T may be obtained from C by writing p for j3s and 

 operating by — S/SiV, and afterwards writing /Ss for p. 



Before putting for the ^4's their values in the determinant (63), it 

 will be well for the sake of symmetry of form, to transform A22 as 

 follows, 



(456) A22 = - (456) (314) (315) (316)Sk/3o, by (67), 



= SkjS2-(314) (316)[(451) (563) - (453) (561)], identically, 

 = S/cj82-[(453) (613) (561) (143) - (451) (613) (563) (143)], 

 = Sk^2- 7^4561(31, 3), by (72), (77) 



where 7'456i(3i, 3) denotes the result of polarizing C456i(p, 3) with 

 respect to /3i and Ps. It is evident that any ' T' which is a function of 

 five vectors only can be similarly transformed. Thus 



(456) Asz = Sk^s ■ 7'4562(i2, 1) (78) 



The determinant (63) may now be written 



Sk02- Ti,6iU 3); Sk^i -04561(2, 3) 



— Sk/So- ^4561(12, 3) 



Sk^1-C456i(3, 2) - SK^r ^4561(13, 2); Sk^s' T ,,,2^, 1) 



(79) 



whose vanishing determines that /3i shall be a double axis, and clearly 

 requires that k shall lie on a quadric cone. The constant C, and its 

 derived constant 'T',' are found at once when the six axes are assigned. 



12. A second method for obtaining a general condition for a mul- 

 tiple axis is to start with (26), which, by (32), becomes 



(456)Fo(p) = A:4^4Ci235(6, p) + /v5/35Ci236(4, p) + h^^Cuui^, p). (80) 



We may make ^4 a double axis by so choosing ki, k^, and k^ that the 

 vector VpF{p), polarized ^^ with respect to 184 and equated to zero, 



12 That is, forming the polar vector as in the note to Art. 11, we write ^a for a 

 after the operation. The easiest way to form the polar vector in this case is 

 to multiply (80) by p and operate by Sp'V- 



