396 HITCHCOCK. 



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

 in terms of k. Thus 



^32 (456) = S/ciSi -(74561(3, 2) - SkIS^ T,,,r{n, 2), (76) 



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

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



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

 will be well for the sake of symmetry of form, to transform ^22 as 

 follows, 



(456) ^22 = - (456) (314) (315) (316)Sk^2, by (67), 



= S/cj82-(314) (316)[(451) (563) - (453) (561)], identically, 

 = S/c/32-[(453) (613) (561) (143) - (451) (613) (563) (143)], 

 = S/cj82-r456i(3i,3), by (72), (77) 



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

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

 five vectors only can be similarly transformed. Thus 



(456) yl33 = 8k^z ■ 7^4562(12, 1) (78) 



The determinant (63) may now be written 



<Sk/32- ^4561(31, 3); (S/ci8i-C456i(2, 3) 



— Sk^o- ^4561(12, 3) 



<S/C|8rC456l(3, 2) — Sk^z- r456l(l3, 2); S/C/Ss- ^4562(12, 1) 



(79) 



whose vanishing determines that /Si 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) = /C4j84Ci235(6, p) + A:5/35Cl236(4, p) + A;6^6Cl234(5, p). (80) 



We may make (34 a double axis by so choosing ki, h-a, 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 fit for p 

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

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



