426 



HITCHCOCK. 



29. Applying this condition for a quadruple axis to (171), the third 

 derivative of pFp in the direction a is 3aFa, because pFp is homoge- 

 neous of degree 3. The other differential operation is the same as 

 SvV-SaV. We may write ^z for a and /Ss/Si for v, and may assume 

 these vectors to form a rectangular unit system, although the homo- 

 geneity of the conditions makes this assumption unnecessary. The 

 operation S/SsjSiV-S/SaV- on VpFp, where Fp has the form (171), 

 yields, (forming the second differential vector and writing 183 for dp 

 and /SsjSi for 5p and j8i for p), 



and VaFa gives SFjSsr. By writing r = w^i -\- gir, and substituting 

 results in (179), we find as the condition that /3i shall be a quadruple 

 axis, 



S^i7r(w^3 + g^) = 0. (181) 



30. It will now be most convenient to distinguish two subcases, 

 according as g is, or is not, zero. If not, we may add to the vector 



Fp a term which will remove from the vector tt its jSi component, 



g 



leaving it parallel to t. We then have Fp as 



IT (a; 1X2 4- gx^) + ^^ixz + iJiX2^ (182) 



The condition for a quadruple axis then appears as 



S/SiTrf = 0. * (183) 



given homogeneous vector Fp. Also, because 7 is the normal to the cones (3), 

 P{oi) = i2(a) = 0. Applying the rule of Art. 18 we find 



ir = aP{y) + yR{y), and t = aP(a, a) + R{a, a) - 2a.Q{a). 



From the relations S^ttt = and g = „ - , we have r = w/3 + g-n-, where 



OApTT 



tt is a scalar and g has the same meaning as in the text. Operating by S-afi 

 gives 



_ Sa^T _ R(a, a) 

 ^ ~ Sa^TT ~ R{y) 



for all homgeneous vector-polynomials, being a triple axis. If we write 

 F{a, a, a) for the third derivative of F in the direction a, and perform the indi- 

 cated operations, we find all terms not of the required form cancel, and 



t' = F{a, a, a) - 3aQ(a, a) - 3g{aPiy, a) + yR{y, a) - aQ{y) - yQ{a) } . 



