346 HITCHCOCK. . 



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

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

 and we have VS/Fp = Vbp; hence the theorem. The converse is 

 equally obvious,^ — 



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

 it can be made irrotational by adding the term pSbp. 



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



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

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

 vector be of the form Vbp. These will appear as equations connecting 

 the nine A'?,, and involving also the axes j8i, ^2, 183. 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 = 02 = as = 0, which is equivalent to neglect- 

 ing the term pSbp. We then operate by FV. Now \/xi-S^i^2^s = 

 — T'i32jS3 and similarly for X2 and X3. Hence easily (using the a's 

 from (4)) 



VVFp-Sl3,^20s = S Kai(a:2T'')3i^2 + xsVjSs^i) (53) 



123 



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

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

 order of terms we may write 



0p = S {VaoV^i^o + T^a3T'/33/3i)Sp/32^3 (54) 



123 



wliich must be of the form Vbp. This is the same as saying that the 

 self-conjugate part of 6 must vanish, or that 6 -\- 6' = 0, or again that 

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

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

 tions; but in tliis case we note that SVOp = 2 S(Fa2F/3ij32-f- VasVjis^i) 



123 



F/32/33= identically, hence the six equations are not independent. 



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

 S^id^i = 0, S^m = 0, SM^z = 0, S^-200s + S^zd^o = 0, S0zd^i + S/3i0/33 

 = 0, and S0id^2 + SjSo^jSi = 0. Since by hypothesis |3i, 02, and (83 are 

 not coplanar these are sufficient to make 6 -\- 6' = 0. By (54) the first 

 and fourth of these equations are 



S- T'^iasT'/Si/So + S^iasV^s^i = (55) 



