448 HITCHCOCK. 



On the one hand we have the vector product of these two expressions. 

 On the other, we have to add to (241) a term which we may write 



— (iSiXi + j82a-2 + 183.^3) {ocxi + yX2 + ZXz), 



where .r, y, and z are undetermined, and are the components of the 

 vector 8. Multiplying out, and equating coefficients of like terms in 

 the variables xi, Xo, Xs, we have a system of eighteen equations. It is 

 clear at the start that we may take a: = 0, for this is the same as 

 saying that /Si must be a zero; which follows from the facts: that /Si 

 is a sextuple axis; and Vtppdp has 3 zeros. Also, we may take pi = 

 qi = n = 0. For this is the same as making the coefficient of Xi^ 

 zero, a necessary condition that /3i be a zero. We then have the 

 system of equations, fifteen in number, 



Coef . of X2^. Coef . of xs^. Coef . of 2:1X2 



1. q\r'- q'r\= 61 4. p"i9"- p'V'i= fif - 2 7. q\r - qr\= - y 



2. v'r\- p'ir'= -y 5. p"r\- p"xr"= 8. pr\- p\r = 



3. p\q'- p'q\= &3 6. q'\r"- q"r'\= 9. p\q - pq\ = a 



Coef. of xiXz Coef. oixzX\ 



10. q\r" - q'r'\ + q'\r' - q"r\ = 13. q'\r - q r'\ = - z 



11. p'r'\ - p\r" + p"r\ - p'\r' = -z 14. pr'\ - p'\r = 



12. p\q"- p'q'\+ V"iq'- pYi= 9i- V 15. p"iq - V q"i = 0. 



Either 4>p or dp may be divided by a scalar which is also multiplied 

 into the other, leaving V4>pdp unchanged. We may therefore without 

 loss of generality assume that some one letter, as r\, is either zero or 

 unity. 



Case 1. Let r\ = 1. I shall show first that p must be zero. 

 For, if not, we have, (by S), p = p\r. Substituting in 14 we have, 

 (since r cannot be zero if p is not zero), p"i = p'ir"i. Then writing 

 for p and for p"i their values in 15 we have, since p'l cannot vanish 

 if p does not, 



r'\q - q'\r = 0. (A) 



Comparing with 13, this gives 3=0. Hence r'\ is not zero, for if so, 

 by 13 and 14, (since r is not zero), p"i = and q'\ = 0, making dp 

 a monomial and V(f>pdp reducible. We may then write r"i = cr, 

 where c is not zero. By (A), q"i = cq and by 14 p"i = cp. From 

 5 and 6 we have 



p"r - pr" = 0, q"r - qr" = 0, 



