QUADRATIC VECTORS. 425 



This result gives, for the second term of (175), the value 



- ^^:^'- (178) 



The remaining term of (174) is similar to (178). For 



g = — , identically, because <Sa(^a = SX --—pFp, 



T(T aha 



as in Art. 18. Again, 



2T<xdTa = - d{(j'') = - 2Sad<7 = - 2S(x4>dp = - 2TaSv(}>dp; 



J rp 



whence = — Sv4>e = — Sv(^a = — Sa<i)v, because 4> is self-con- 



dh^ 



jugate. Thus the two terms of (174) are together equal to 



which is in convenient form for differentiation. If /3i is a triple axis, 

 g is independent of X. If /3i is to be a quadruple axis, it is necessary 

 and sufficient that (179) shall be independent of X. The reasoning 

 by which this condition has been obtained is independent of the degree 

 of the given vector Fp, and applies, therefore, to vectors of any degree. 

 The denominator Ta, as already shown, takes the form SXjStt, when 

 /3i is written for p. It can easily be shown that, for all homogeneous 

 vectors, the expression (179) takes the form 



where r' is a new vector.^'' The condition that this fraction shall be 

 independent of X is S^ttt' = 0. 



17 To prove this in general, we may write Fp = aP + ^Q + yli, where a, /?, 

 and 7 form a rectangular unit system as in the text, and P, Q, R, are homogene- 

 ous scalar polynomials of degree n in x, y, z. If we write d and 5 for two inde- 

 pendent symbols of differentiation; and put dP — P(5p), 8P = P{Bp), and 

 d8P = 8dP = P{dp, op), it being understood that ^ is always put for p after 

 the differentiation; with a similar notation for the first and second differentials 

 of Q and of R; we must have P0 = Qff = R0 = 0, because /8 is a zero of the 



