8 Proceedings of the Royal Irish Academy. 



If these are both zero, we shall therefore have simultaneously 



S y 0~ Y6a<pfi = and Sy$' Vdaffi = 0. 



But 7 was an arbitrary vector, hence these equations imply 



d'V6a(j>ti = and $>' F0a<£/3 = 0. 



that is, the vector TQa<pfi is a zero for both of the conjugates <£' and 0'. 

 This direction will then be a zero for the conjugate of any function of the 

 pencil. 



The two sub-cases may accordingly be characterized as follows : — 



1. <p and 6 have a common /em. This implies that the conjugates <j>'p 

 and 6'f> lie in a common fixed plane for all values of p . 



2. <p' and 0' have a common zero. This implies that ipp and dp lie in 

 a common fixed plane for all values of p. 



The second case is easy to formulate. Since the common fixed plane is 

 known when q> and 6 are known, we may choose i and j any two perpen- 

 dicular unit vectors in that plane, and expand q, and in terms of them, when 

 the two functions necessarily appear in the form 



<p P = iSa lf , - jSarjp, Bp = iSfiXp + jSfipp, (13) 



when- £. ij. A. and u are four constant vectors; for, by hypothesis, <pa = and 

 0/3 = 0. To find t w< o = - Sptp'i = SpVat, whence, p being any 



vector, we must have Vat = - f'»; therefore t may be any vector at right 

 angles to j'i, distinct in direction from n, the tensors of a and of * being 

 selected t" satisfy this equation. In a similar mannei we may find values 

 fur 77, X, a- 



In the other sub-case f'p and 8'p lie in a common fixed plane. Hence 

 they must lie. in form, the gates of the right members of (13) or some 



equivalent That rite for this case 



+p = FaeSip - VnnSljp, e r> = V$\Sip - VfinSjp, (14) 



where t, ij. A. and p. ai found by treating the conjugates as <f> and 6 



were treated in (13), < andj are taken in the plane of the conjugates, k is the 

 common zero of <f> and 6, and a and /3 are the directions annulled by the 

 conjugates, <f>' and ff, respectively. 



The forms (13) and (14), from their method of formation, cover all 

 function-pairs such that no function of the pencil has an inverse, 



10. Application i . — In conclusion I shall pi 



