100 
Proceedings of Boyal Society of Edinburgh. [sess. 
been, as, although there are two general integers involved, viz., p 
and n , Schweins attends only to the second of them. He begins 
with the case of n = 4, p = 2, — that is to say, the multiplication of 
\a r b s dd u \ by %a h b h , 
the result being 
(A' 
a 2 . «3 . «4 
h h jh \(2) 
)= 
flA? +ai A* 2 + “ 2 AX 4 ) 
+ |Aj +<ll A>fX 1 ) 
+ ||Ai +<1i aX 8A 4 + “ 4 ) 
+ |AfA^ + X + X 4 ) 
+|a>^ +<,2 aX + “ 4 ) 
To indicate the mode of formation of the alternants on the 
right from the given alternant on the left, he says : — 
“Hier entstehen alle Vertheilungen von h, h zu zweien in 
vier Abtheilungen, namlich 
Jl -p CL-^ j 
ll + <^2 
a 3 i 
a 4 
ll -p Cl] 
Q>2 
ll + <X 3 
a 4 
h+ct] 
a 2 
a 3 
h + a 4 
a x 
li + a 2 
\ _ 
h + a 3 
a 4 
a x 
h +a 2 
«8 
h + ci 4 
a \ 
a 2 
h + a 3 
h + a 4 
He next takes the case where n = 5 and p = 3 : that is to say, 
the case of 
\a r b s dd n e v \ . %ci ]l b h c h , 
