THE AXES OF A QUADRATIC VECTOR. 347 



(56) 

 and the others are obtained by advancing subscripts. 



Inspection of the scalai* products which occur in these equations 

 shows that irrotationahty of the quadratic vector Fp is dependent 

 on the relation of the a's to the two systems of vectors (3i, /So, jSs and 

 ^j32i33, FjSsiSi, T'(3i/32. The most natural procedure is to expand the 

 latter svstem thus 



T%/33 -5^1^2/33 = ^11^1 + hilSo + &31/33] 



T'^3^r Si3i52^3 = &12^1 + M2 + ^32^3 V (57) 



T'^|8ii82- S/3i/32i33 = bn^i + 623^2 + &33)33 j 



whence we have 



*ii = S-J%8zVl32^3 = S%3s - iSl^l; bo, = hi. = S-]%l3sV^z^i 



- /3^S/3u82 - S/32i33S^3^i (57) 



and similarly for the other 6's. We may now introduce the expan- 

 sions of the a's from (4) and of V^o^z etc. from (57) and the six equa- 

 tions of form (55) and (56) become 



^33^22 — ^22^ 33 + &23(-423 — ^32) =0 (58l) 



fell^33 - ^33^11 + &3l(^31 — ^13) = (582) 



622^11 — ^11^422 + 612(^12 — Aoi) = (583) 



- feu(^23 - ^32) + 6i2(.43i + Ars) - bn{An + ^21) = (584) 



- 622(^31 - ^13) + &23(^4i2 + .421) - 612(^23 + ^32) = (585) 



- &33(^4i2 - A21) + bisiAn + ^32) - 623(^431 + ^13) = (580) 



Here we note that if the six equations be multipled, in order, by 

 hn, boz, bzz, 623, &31, &12, and the results added, the sum of the left 

 members is identically zero; the six equations are not independent. 



Since the yl's have already been determined as functions of the axes, 

 the six equations (58) are necessary conditions which the axes must 

 satisfy when the quadratic vector is irrotational. 



We note further that, assuming jSi, 182, and jSs to be real, the scalars 

 &11, 622, and ^33 are different from zero. These three scalars could all 

 vanish only if V^-i^z, V^h^u and V^i^-i were all minimal vectors, i. e. 

 imaginaries of null tensor; thus in general we may assume 611 different 

 from zero; and except in this very special case, therefore, (58i) is a 

 consequence of the other five equations which, with the exception 

 noted, are sufficient that the curl shall be of the form Vbp. 



