THE AXES OF A QUADRATIC VECTOR. 345' 



Case 3°. Only one of the o's is zero. Suppose 03= 0, with ai and 

 02 not zero. Then pi= ^2= 0. The six equations reduce to 



«2 + .I31 = 



Ol + .4 32 = 



^33 = 



Hence if A33 = the a's can be determined and a coplanar set exists. 

 Similarly for Aw and ^22, in agreement wath a former result. 



Summary of tests for coplajiarity of axes. 



The quadratic vector (31) possesses three axes in the same plane when, 

 and only when, one of these seven condiiions holds: the vectors ai, 02, o-z 

 defined by (4) are coplanar; one of the three determinants of the form (49) 

 vanishes; or one of the constants An, A22, ^33 vanishes. 



10. Application to Irrotational Vectors. 



As a second illustration of the utility of the constants An etc., I 

 propose to examine the properties of a quadratic vector under the 

 requirement that it be irrotational, i.e. its " curl " shall be zero or 



VVFp = (50) 



The significance of this equation on the physical side is well known. 

 To see the algebraic aspect of the problem we may recall that, as was 

 shown by Hamilton,^ any linear vector function cpp may be written 

 as the sum of two terms thus. 



<PP 



cop -{- Vep (51) 



where e is a vector; co is self-conjugate, irrotational, and has its three 

 axes at right angles to each other. It is natural to enquire what 

 restriction is imposed on the axes in the case of an irrotational quad- 

 ratic vector. 



The scope of the enquiry will appear from the follo^\'ing: 

 Theorem 8. If a quadratic vector can be made irrotational without 

 altering its axes, its curl is of the form V8p. 

 The proof is evident from the identity 



rv{Fp + pS8p) = WFp + Vp8 (52) 



7 Elements, Art. 349; 2nd Ed. p. 492. 



