4 Proceedings oj the Royal Irish Academy. 



so that if we agree to write 



cSafifii = Cj and .</ + ciSafifit = ffi, 



we may rewrite in terms of its effect on o, /3, and /3i, thus — 



0/> . Saj3/3, = (<A + C 2J u)Sj3/3 1/0 + <//iS/8,ap + ^."Sn/Sp, (50 



where the analogy with (1) is more obvious, g and g, being the two roots of 

 the cubic in <p~*d. To complete the formulation, put 



a 2 Sa/3/3, = F3/3„ /S=S«f3/J, = F/3,a, y : Sa/3/3, = Fo/3, 

 whence (5) and (3 2 ) become 



<pp = \Sa z p + /u/?/i./> + vSy.p, | 



fyj = v\ - (';ju)6'a 2 p + gpSfiip + g,vSyip j 



It is clear that the form of </> is as in (1 . while differs only in its effect on a. 



4. Geometriec 'ti<m. — It equations of a curve are given as in (3), 



we write down the conditions for self-conjugation from (.") and (5,) — 



FAF/3/3, - r„r,-j.<, - VvVafi = 0, PqiFflS, + Vc lV Va(S = 0, 



which (by Hamilton's £ equivalent to the six scalar 



equations 



<S/9,/i = S/3i Sav = <S|3,A, S(3\ = 6'a^, 



c,5j3» = 0, e,6W = cS^./u, eS/3/i = 0. 



To solve, regarding a, /3. and /3, is known vectors, we note Brat that c and c, 

 are by hypothesis different from zero. The six equations thus reduce 

 at once to 



S/3 M = 0, Sj3,n - 0; Sav = 0, 6'/3k = ; S/3.A = 0, and ,S'/3A = Sap 

 Hence ji, being perpendicular to both $ and /9i, has the form ^F/3/3, where 

 /- is some constant - Similarly n has the form '/Fa/3, and A has the 



form rF/3/3i +;>J 7 /3,a, where q and r are two more constant scalars. Collecting 

 results, (5) and (5,) be< 



4>p. 8afl8, = (rF/3/3, - pF/3 l a)S^/3,p+pFp/3,.-S'/3 I ap - ? FajB . Sa/3p, ] (? 

 dp = g+p + cpVpPiS(ip,p iSafjp. I 



The self-conjugate character of these two linear vector functions is evident 

 from their form. And since, in the geometrical problem, any multiple of one 

 of the quadratic expressions (3) may be added to the other without altering 

 the curve of intersection oi the two quadric surfaces, we may take // = 

 in (7). If we now opei 7. bj Sp and introduce a set of oblique 



coordinates defined by the equations 



*6'a/3/3i = Sfijiip. Sa£ >,-■•"> stfa/3/3i iSajtJp, 



