Hitchcock — A Study of the Vector Product V(f)a0fi. 31 



2. Proof that V<l>a0(5 t- V(ia<pji is a , function of F(i/3. 



The main proposition follows at once from Hamilton's definition of the 



cvnjufjatc of a linear vector function : if ^ be any such function, its conjugate 

 \f/' satisfies the relation 



Sfjxpa = Saip' f), (4) 



where f) and a are any two vectors. 



Consider now Iho term V(i>u0f5. Since it is linear in ji, we may take p 

 any vector whatever, and transform as follows : — 



SpV.pa6ii = - S<paVpei5 



= - Sa(p' Vptiii, as in (4), 

 = - Saxpfi, say, (5) 



if we agree to write 



^Pfi^f'Vpdii. (6) 



Again, starting with the term VOafIS, we may write 



Sp F6»a(/,/3 = + S<j>(5 VpOa 



= + Slicf>' Vpt)a, as in (4), 



= + Sjii^a, by (6J, 



= + Sc4'(5, by (4). (7) 



Adding the two results (6) and (7) gives 



Sp[ V,paefi + Vea<j,t5] = - Sa[^P - ^'J/S. (8) 



But it is well known that an expression of the form [ip - \p']f5 is of the 

 form Vij3, where £ is a vector ; and, since the left side of (8) is linear in p, 

 the same must be true of the right side, whence we may write 



[i - f ] /3 = V^p(5. (9) 



where np (which is the same as t) is a linear vector function of p. The 

 equation (8) now becomes 



Sp[V,j,a9li + V9a^^] = - SaVnpii 



= + Sllf^TTp 



= +6p7r'Faj3; (10) 



but, since p is any vector whatever, it follows that 



V(Paei5 + Vea,j>l5 = tt' F(./3, (11) 



which shows that the left side is a linear vector function of FVi/3. It remains 



to study the form of tt'. 



[4*] 



