426 



HITCHCOCK. 



29. x\pplying 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 183 for a and jSs/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/Ss/SiV • SjSsV • on VpFp, where Fp has the form (171), 

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

 and iSsjSi for bp and |3i for p), 



r/33x + F^if, 



and VaFa gives SFjSar. By writing r = iv^i + gir, and substituting 

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

 axis, 



Sj8i7r(w)/33 + 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 j8i component, 



9 

 leaving it parallel to t. We then have Fp as 



7r(a;ia-2 + g^s"^) + ^^2X3 + /xx^^ (182) 



The condition for a quadruple axis then appears as 



S/3i7rf = 0. (183) 



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

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



TT = aP{y) + yRiy), and r = aP{a, a) + R{a, a) - 2aQ(a). 



From the relations S^itt = and g = „^ ,, , we have t = m/3 + gir, where 



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

 gives 



_ Sa0T _ R(a, a) ' ' ' 



^ ~ Sa^ir ~ 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 



r' = F{a, a, a) - 3aQ{a, a) - 3g{aP{y, a) + 7^(7, a) - aQiy) - yQ(a) } . 



