THE AXES OF A QUADRATIC VECTOR. 347 



(56) 

 and the others are obtained by advancing subscripts. 



Inspection of the scalar 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 tu'o systems of vectors |3i, ^2, 183 and 

 V^ojSs, VjSsSi, T'/3i/32. The most natural procedure is to expand the 

 latter system thus 



r,32i33-Si3i/3o,33 - bn^i + ^21^2 + 631133 1 



F^3/3i -5/3^^3 = 612^1 + M2 + hrM^ V (57) 



Fi8l^2 -5/31132/33 = 6l3/3i + />23^2 + /^33^3 ) 



whence we have 



*ii = S-r/32/33T'/32)33 = S%8i - ^l^l; bn = bn = S-J%^sVl3sPi 



= ^IS^i02 - S/32/33S^3i3i (57) 



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

 sions of the a's from (4) and of V^2^s etc. from (57) and the six equa- 

 tions of form (55) and (56) become 



^33^22 — ^22^33 + t23(--l23 — -^32) = (58l) 



&11^33 - &33^11 + 631(^31 - ^^13) = (582) 



622^11 — 611^22 + &12(--ll2 — --^21) = (583) 



- bn{An - A^2) + 612(^31 + ^13) - ^>i3(.4i2 + ^21) = (584) 



- 622(^31 - ^13) + 623(^12 + ^21) - bi2{A2^ + .432) = (585) 



- &33(-4i2 - ^l2l) + &13(^23 + ^32) " ?>23(^431 + An) = (583) 



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

 bn, b22, bss, 623, 631, 612, and the results added, the sum of the left 

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



Since the ^4'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 /3i, ^2, and |33 to be real, the scalars 

 bn, 622, and 633 are different from zero. These three scalars could all 

 vanish only if Vj32^s, T'/35/3i, and T'/3ijS2 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 T'5p. 



