525 
1907-8.] Algebra after Hamilton, or Multenions, § 5. 
(where — .... a a ,^b = /8 2 /3 3 •• • A>)> an( l we h av e to show that r 
is zero. To show this we prove that always 
®a+c®c^a* ^b+c^b^c P (®^y ) 
consists of (a + 6) th and lower order parts ; and therefore in particular that 
&a+<^c a l^a-l‘^b+c^b—l^c a l 
consists of (a + b — 2) th and lower order parts; and therefore that r is zero. 
P = (^a+c^c^a^ a+b+c&cJZb&c) (^a+ b+ffi afi b^ cf>b+J3 ^ • 
Here by (7) the first bracket is of order b and the second of order a. 
Hence p consists of (n + 5) th and lower order parts. 
Now by linear variations of the above six types the a-\-b + c fictors 
cq . . . . a a /3 1 .... /3 b y 1 . ... y c can be reduced to a set of fictits (or rather 
we can make cq = x x i v . . . . , y c — x a+b+c i a+b+c ), and when this has been done 
the third and fourth forms of (10) each equals 
^c^cPb^c- 
Exactly similar reasoning, step by step, applies to the case when 
the fictors TU a m b m c are not independent. [If m a ts c are not independent, 
or if m b Tjj c are not independent, (10) is obviously true. When zn a z u c are 
independent, and zj5 b zz c are independent but zu a zs b Z!5 C are not independent, 
then some one f3 at any rate can be expressed in terms of the other 
a+b+c— 1 fictors.] 
In the equation of the last two forms of (10) first change zn a to zj5js x> 
and again in the same equation change U5 b to zj5 x zs b . We thus get two 
equivalent forms of 
^a+b+x(fia+b+c+offi b' 
namely 
^a+b+xifia+c+s^ x- ^5+c^b^c) ^a+b+x\fia+ffi a’^b+c+offi x^b^ c) ■ (H) 
which shows that in such transferences any product U5 X may be transferred 
provided it is not the explicitly common product zs c of the two com- 
binatorial parts. 
(11) , again, may be modified by changing ZJ3 c z3 a to za a and zz b zz c to zz b and 
remembering that the number c is now defined by saying that there are 
c fictors which are common to each of the sets, zzr a of a fictors, and m b of 
b fictors. Thus (11) becomes 
®a+&+a:— 2c(®a+»^a^x* = ®cH-6+*— 2c(®a^a*®6+a:^flc^&) • * • (12) 
(12) appears to me to contain all the essential features of both Grass- 
mann’s progressive (c = 0) and regressive (c > 0) multiplications. 
We frequently meet with expressions very analogous to quaternion 
