Hitchcock — On the Formulation of two Linear Vector Functions. 7 



equivalent to the three scalar equations 



', N/3/3, = 0, c.&ift - oSj|3/3, = 0, c/3- - 0. 



Tf c and c, are different from zero, these reduce to 

 S/3/3, - 0, ,S'«/3, = 0, j3 2 = 0, 

 whence /3 must be a vector whose square is zero, i.e. a minimal vector. An 

 example of such a vector is i+j\/~ 1. We may satisfy the conditions 

 by taking |3i = k, and a = i, with c and Ci any constants. We shall then 

 have 



<pp = 9? + c (' + ]•/ ~ l ) s (~ J + *•/ - 1)p + cJeSkp. 

 Here t, _/, & are any three unit vectors forming a rectangular system, <f> is 

 self-conjugate, and has the two axes k and i + j \/ ' - 1, the latter being 

 a double axis. 



When <pp has been developed in this or any other manner, a second 

 function dp may be formulated, following Joly, by expanding in terms of 

 three vector constituents of <p. Thereby 6 is determined by means of nine 

 constants, and it is in terms of these constants that Joly expresses a large 

 number of invariants (cf. note 1). 



9. Typical form of <p and when no function of the pencil has an inverse. — 

 The function-pairs (1), (6), (10), and (12) include all possible cases except 

 when no function of the pencil possesses an inverse. In this latter case, the 

 cubic in <j> + t6 must have a vanishing root for all values of t. Now the 

 constant term in this cubic, as was pointed out by Joly, may be written 



riii + tl 3 ' + t% + t 3 m- it 



where m/ and m a are the third invariants of ty and respectively, and h' 

 and li are Joly's new invariants given by 



_ S(<p a e j30y + (j>f36y6a + (pytiaOfi) , _ S(Qa<l>\5<py + 0fi<pyrj,a + tiy^affi) 



Sajiy Safiy 



The invariants tin and m 3 vanish by hypothesis. If, and only if, // and 

 l 3 also vanish, we shall have no function of the pencil possessing an inverse. 



These quantities are invariant in the sense that their values are inde- 

 pendent of our choice of the three vectors a, /3, 7, provided they are diplanar. 

 Let a be chosen to be the direction annulled by <p. that is r/>u = 0. We 

 shall have two sub-cases, according as 0a is, or is not, zero. If da = 0, it 

 is evident that a is zero for every function of the pencil. If da is not zero, 

 let j3 be chosen to be the direction annulled by d, i.e. 0/3 = 0. Let y be any 

 vector not coplanar with a and /3. Joly's invariants now become 



, _ Sj,fi6yda , Sda-p^y 



Safiy iSafiy 



