574 Proceedings of the Royal Society of Edinburgh. [Sess. 
of (p is 1], so that log 0 (p is i/-skew, and therefore (p= e 10ff o^is in the desired 
form e*. 
If we try to put <p = x y i r where yfr is i/-self -conjugate and x is r-rotational 
we must have 
p'<p = ipyx^ = «A 2 • 
Hence if cp'cp has no square root (and such cases do occur), the resolution is 
impossible. If, however, <p~ l is not infinite, we may put \[s = J 0 ( <p K cp ) ; and 
this form of \[r is y-self -conjugate by (32) ; and now we have 
X = <t>'1'~ 1 
is necessarily ^-rotational, since we have 
XX = V - <£<U 1 <£ W1< £' = 1 • 
Thus when 0 _1 is not infinite we have a solution of the problem, but there 
are always many solutions if there is a single one. Under the same circum- 
stances the resolution <p = \p~x is possible. We get \/s = J 0 (cp(p'), x = y i r ~ l( t>- 
I have found, by an argument too long to reproduce here, standard 
forms for a y-skew and for a y-self-conjugate linity which reduce to those 
considered at the end of § 7 when v — 1 and <p is real. 
It is not difficult to show that invariably the \ v X 2 . . . of (22) § 7 
may be so taken that 
X x = aqAj, X 2 = x 2 \ 2 , . ... p l = Z/i/^u • • • • 
where every x, y .... is either unity or zero. [Think of the fictorplex of 
independent fictors \ g , ... . which survive the operator ; take 
\ = Zo^Kr ^2 = • • • • 
\ 1 U 5 * • • • 1 
then think of the fictorplex not included in the above, which survives 
£ 0 g ~ 2 , and so on.] £ £ 0 , rj, rj 0 , ... . are thus subdivided into parts which 
themselves satisfy all the conditions of (20), (21) above. But \ v \ mJ 
still remain to a certain extent arbitrary. This arbitrariness is further 
diminished in the standard forms referred to. In the following enuncia- 
tion, g is the number of fictors in one of these subdivisions, and not the 
original value, in general, of the minimum degree. Similarly, £ and 
refer to such a subdivision instead of to the sum of the subdivisions. 
When <p' = do <p we may put 
ip = (afiplvfig ± PgSpp-'aP) 4- (aSp\vPg-i ± S/o|v -1 a 2 ) + . . . . \ 
ioP = (aiSp|v^_i ± fig-fip^ 1 ^) + {a 2 Sp\vP g _ 2 ± ft_ 2 Sp|j/“ 1 a 2 ) + . . . • L (33) 
where d 1 = vf3 g , fig = v~ l a 1} a 2 = v^_ x , v -1 a 2> .... J 
where 8 V /3 2 , ... . are the same as a v a 2 , .... in two cases only ; namely 
