522 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of order n. We have just seen that this may be taken as the multiplex 
i 1 i 2 . . . . i n where i v t 2 , ... . are fictits. Let p a , p b be multenions given by 
Pa = 9a + 9a - 2 + 9a-t + Pb = Zb + 9b- 2 "+•••• 
Suppose all these symbols expressed in their simplest extended forms 
in terms of the multits, and think of the process of picking out the various 
parts of, say, q a q b when n > a-\-b. It is clear that the highest order parts 
that can occur are of order a + b. Lower order parts are in general included, 
but they only arise by two or four, etc. fictits, disappearing by fusion with 
scalars on account of the relations if = scalar, etc. It follows that the 
product p a p h p c (of, say, 3, though the reasoning is general) is of the form 
9 a+b+c 4" 9 a+b+c — 2 4" 9 a+b+c — 4 4" • • • • 
Thus in S a+b+c p a p b p c we may change p to the q with the same suffix; 
and S k p a p b p c = 0 whenever k exceeds a + b + c or differs from a + b + c by an 
odd integer. 
Now the fictor products U5 a , xa b , t?j given by 
&a = a +2 ••••<*«> = /hA> • ■ • • Pb, ®c = Y 172 • * * -Ye • • (!) 
are of this nature (i.e. js a —p a = 9a + q a -2 + • • • •) as we see by expressing 
each fictor in terms of fictits and each fictor product in terms of multits. 
Hence we have 
&a&b * * • • = 9a+b+.... + 9a+b+....-2 + 9 a+b+....-i + • (2) 
Hence 
^a+5+. . . J^aJ^b • • • • ®a+6+.... ([SJ®.)([S»]®.> (3) 
where the square brackets imply that we may retain or reject the symbol 
enclosed at will. Also 
1 0 . . . . . ( 4 ) 
whenever k exceeds a + &+... . or differs from a + 6+ . . . . by an odd 
integer. 
S a tu a is a combinatorial part of the fictor product trr a . [This means 
fundamentally that if two consecutive fictors a, a be interchanged the 
part alters in sign, but otherwise remains unchanged ; derivatively it means 
that if any two a, a be interchanged the sign thus alters, and that the 
part S a z*y a is or is not zero according as cq, a 2 , . . . . a a are not or are 
independent.] S a t* 7 a is indeed Grassmann’s progressive product of the fictors 
when a does not exceed n. [When a>n, a x a 2 . ... a a cannot be inde- 
pendent, so that then = 0. S a ttJ a is therefore not Grassmann’s 
regressive product.] 
