6 Proceedings of the Ro>/>tl Irish Academy. 



where p, q. r are three arbitrary constants. If we substitute these values of 

 A. u, v, in (9) and (9,), the self-conjugate character of <p and is evident. 



7. Typical form of <f> " >Jun <f>~ l has an infinite number of axes. — It 



may also happen that <p~0. whether possessing a double axis or not, has an 

 infinite number of axes. The most general form of <f l 9 is then* 



<f>' l 8p = up ~ yStp. (11) 



where g is a double root of the cubic in <p~'0 (not necessarily corresponding to 

 a double axis), and y and i are constant vectors. Lot a and /3 be two vectors 

 which, with y. form a diplanar system. Call fa, <pfi, and >uy, as before, A, p. 

 and !•. respectively. Expanding tpp we have 



yp . Sajfiy = XSrfyp + p.Syap - vSa I (12) 



and by acting on (11) 1 



Bp = >/4>u - . ■ ( 1 . 



same way the direction of 7, and also 

 that of any vector perpendicular t«> 1. If y is rpendicular to t, the 



function ^"'6 becomes a i y l>ecomcs a double 



If we require <p and 6 t . we may take f any self- 



conjngate linear vector function whatever, and must have t parallel to v. 

 Operati; _ Sp, mil setting g = ii _ we see that SpOp is a perfect 

 c may be - from the'pemil. 



We may, if we w . <1 various special (12) and fl2i) as being 



merely limiting forms of the function-] dii ami (10). For example, 



if either c or '-, vanishes, (8) falls into the form (11), and may be written as 

 in (12). 



:iction-pairs have 

 been built up by assigning " Various interesting 



special cases obtained by assigning .1 form to <f>. As an 



illustration, important in the th- Jtant torsion, let <p l>e 



required t«. have a double '*• self-conjugate. Ii is clear that <p 



cannot be real, since a 1 . linear vector function has tl 



distinct, mutually perpendicu: :i we write tf> in the form of 



the right member of (4), whi g neral form of a linear vector 



function having two coincident . : — 



<t>P -p. 



the condition for self-conjugation is 



• Loc. at., p. 175, e4aatK.11 I.- 



