528 
Proceedings of the Royal Society of Edinburgh. [Sess. 
be similarly modified by passing m ( = product of the n fictits) into each 
S n ( ), thus 
qSuJa ic) a {n - c) = % 2 a { c c ^q a { ^ ] . . . ( 3 ) 
To prove (1), first suppose .... a_ n are independent. In the proof 
it is convenient to say that a {c) and a (n ~ c) are supplementary products. 
Suppose a (c) and f3 (e) are two of the products. Then 
= 0 unless = ±aj?_r c c) . . . (4) 
for, by the reasoning of § 5, if a (c] and /3 (e) contain no common factor, 
a {c) /3 (e) consists of (c + e) th and lower order parts, so that unless /3 {e) is supple- 
mentary to a (c \ S^a^/3? = 0 ; and if a {c) and /3 (e) contain any common factor 
of a fictors, a (c) ,8 (e) certainly contains no higher part than the ( n — a) th [and 
not so high a part unless all n fictors are present in « (c) and 8 {e) ]. 
Also (when a_ l5 a_ 2 , .... are independent) the T values of af are inde- 
pendent. For suppose, if possible, 
xaU + yf3? + . . . . = 0. 
Operating by S n a£r c c, ( ) we get x = 0. 
We may then put q in the form 'Ifxaf and also in the form 2 2 yajT_r c A 
Operating on these two forms by S„( )a?r c c) and S w a?( ) respectively we 
get the two forms of (1). 
It will be noticed that we may definitely state, consistently with the 
above, 
a (n) = a (c) a (n - c) = a_ x a _ 2 .... a_ n , a (0) - 1 . . . (5) 
When «_!, a_ 2 . . . . , a_ n are not independent first suppose a_ v a_ 2 , . . . . , 
a_ (n _D are independent, and let 8 be a fictor of the fictorplex which is. 
independent of these n— 1 fictors. Thus (1) is true when for a n we write 
either a n + /3 or a n — f3. Adding, (1) follows for this case. We may now 
suppose a_i, a_ 2 , .... a_ ( „_ 2 ) alone to be independent, and so on. 
(1) is, of course, a generalisation of the two well-known quaternion 
theorems, 
pSa/5y = aS/3yp 4- BSyap + ySa (3p 
= Y fiySap + YyaS ftp + Va/3Syp. 
When I started to search for the theorem in our present subject 
corresponding to these, I confess I scarcely hoped to find so general a 
result. When it emerged I at once discarded my previous modes of 
groping after the key to the present methods, and selected the path of 
“ orders ” and of “ parts.” The great powers of the quaternion K q in the 
fundamental logic and symbolism of quaternions had previously attracted 
and, I cannot help thinking, misled me. 
