THE AXES OF A QUADEATIC VECTOR. 345- 



Case 3°. Only one of the as is zero. Suppose 0.3= 0, with oi and 

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



flo -h Au — 



Azs = 



Hence if ^433 = the a's can be determined and a coplanar set exists. 

 Similarly for ^u and ^422, in agreement wath a former result. 



Summary of tests for coplanarity of axes. 



The quadratic rector (31) j^ossesses three axes in the same plane when, 

 and only when, one of these seven conditions holds: the vectors ai, ao, as. 

 defined by (4) cii'& coplanar; one of the three determinants of the form {49) 

 vanishes; or one of the constants Aw, A22, Azs 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 



VS7Fp = (50) 



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

 To see the algebraic aspect of the problem w'e may recall that, as was 

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

 as the sum of two terms thus, 



(^p = cop + T'ep (51) 



where e is a vector; 00 is self-conjugate, irrotational, and has its three 

 axes at right angles to each other. It is natural to enc^uire what 

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

 ratic vector. 



The scope of the enciuiry will appear from the following : 

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

 altering its axes, its curl is of the form 1^8 p. 

 The proof is evident from the identity 



T^V(Fp + pS8p) = WFp+Vpd' (52) 



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



