Hitchcock — On the Formulation of two Linear Vector Functions. '■> 



'2. Possibility of the reduction to form (It.— I \i all these problems we are 

 concerned with the properties of a pencil of functions <j> + td. That is to 

 say, in studying the curve (3), or in forming the vector product V<l>pOp, or 

 in calculating Joly's invariants, we may replace a given <j> or by any other 

 functions of the pencil. The general problem is, then, to determine what 

 types of function-pairs may occur. 



In writing <j> and in the form (1), Joly assumes the possibility of the 

 reduction. To see on what the possibility depends we note, first, that Vfiy, 

 Vya, and Fn/3 are the axes of $"'#.' A necessary condition for reduction to 

 the form (1) is, therefore, the existence of three distinct axes for <p' l 0. 



Second, if $ has a vanishing root, so that <£-'# does not exist, we may 

 suppose replaced by a different function of the pencil. But it may happen 

 that no fuuetion of the pencil possesses an inverse. The reduction (1) is 

 impossible in this case also. We have therefore to examine two cases : — 



I. The function f~'d exists, but has a double or a triple axis. 



II. The function r/r'fl cannot exist, i.e. no function of the pencil + tO 

 possesses an inverse. 



3. Typical form of <j> and 9 when <p~ i 9 has a double axis. Suppose r/T'fl to 

 have a double root. As I have shown in a former paper,* we may then write, 

 as the most general form of <i>- l 6, 



<p->0 = gp + c/3S/3/3,,o + cfi x Safi P , (4; 



where tj is the double root of the cubic in <j>~ l 8, and may vanish. 



The double axis is /3. It is assumed that c, c,, and Safifi, are all different 

 from zero. To express <f> and 6 in a simple manner we have now merely to 

 operate by <j> on the three diplauar vectors a, /3, /3 t . Let the results be 

 denoted by X, p, v, respectively ; and these three vectors are also diplanar, 

 because, by hypothesis, <j> has an inverse. Expanding <j> in terms of A, /u, i> we 

 have 



(j>p . SafijSi = \Sfifi,p + fiSfijaft + vSafip, (5) 



and by operating with rj, on both sides of (4) we have 



dp = r/(pp + C/*/Sj3/3,/3 + Cii'SajSp (5i) 



These expressions actually differ little from (1). To bring out the analogy 

 we first note that (5,) is equivalent to the three equations 



0« = y\ + Pi uS«/3i3„ 0/3 = //,,, 6(3, = (g + c^fr)*, 



*Pioc. Koyal Sioc. Edinburgh, vul. xxxv, Part II (No. 17, June, 1915), p. 172. 



[1*] 



