34G HITCHCOCK. 



for if Fp can be made irrotational by adding pS8p, (the only form of 

 term wliieli leaves the axes unaltered), the right member must vanish, 

 and we have WFp = V8p; hence the theorem. The converse is 

 equally obvious, — 



Theorem 9. If a quadratic vector has its curl of the form V8p 

 it can be made irrotational by adding the term pSdp. 



11. Conditions that the Curl shall be of the Form V8p. 



I shall now show that, in general, a set of five scalar equations 

 exists wliich are necessary and sufficient that the curl of a quadratic 

 vector be of the form T'6p. These will appear as equations connecting 

 the nine A's, and involving also the axes /3i, /S2, ^3. Since the ^'s 

 have been obtained as functions of the axes, these equations impose 

 restrictions on the axes of an irrotational quadratic vector. 



Taking (31) we let oi = a^ = 03 = 0, wliich is equivalent to neglect- 

 ing the term pS8p. "VYe then operate by T'V. Now \/xi-S^i^2^z = 

 — T'jSsiSs and similarly for .T2 and X3. Hence easily (using the a's 

 from (4)) 



T'VFp- S^i/32^3 = 2 Vai{x2V^^2 + xsl^l^y) (53) 



123 



The right side of this equation is a linear vector function of p, which 

 we may call Op. Putting for the x's their values, and arranging the 

 order of terms we may write 



ep = Z (raor/3i/32 + T'a3T'/33/3i)Sp/32^3 (54) 



123 



wliich must be of the form T'6p. This is the same as saving that the 

 self-conjugate part of 9 must vanish, or that + 0' = 0, or again that 

 Spdp must vanish for all values of p. In general the vanishing of a 

 self-conjugate linear vector function is equivalent to six scalar equa- 

 tions; but in this case we note that S\/6p = 2 S(T'a2l'/3i/32+ T'asT'/Ss/Si) 



123 



V0205= identically, hence the six equations are not independent. 



A simple way to set up the six equations in explicit form is 

 Si3i0/3i = 0, S/320^2 = 0, S/33^/33 = 0, SjS-.d^s + SMI32 = 0, S0^^i + S^id03 

 = 0, and SdidjSo + S^2dl3i - 0. Since by hypothesis /3i, 02, and ^3 are 

 not coplanar these are sufficient to make d -\- d' ^ 0. By (54) the first 

 and fourth of these equations are 



S- T'^ia2T'/3u82 + S|3ia3r/33/3i = (55) 



