38 
Proceedings of the Royal Society of Edinburgh. [Sess. 
variables be common to three rows , or one variable common to four rows , 
the determinant vanishes. These are easily established. The second 
only of them deserves a word of comment, its intrinsic interest being 
enhanced a little if the factors be removed in a particular order and the 
co-factors at each stage of the process have their affinity of form brought 
to light. Thus, taking the corresponding property in the case of the five- 
line determinant 
[~ abed 
b c d p 
c d q r 
d s t u 
v w x y _ , 
the factor p — a can be removed and the first two rows left in the form 
1 b + c + d bc + bd + cd bed. 
1 b + c + d bc + bd + cd bed ; 
the factors q — b,r — b can then be removed and the first three rows left 
in the form 
1 c + d cd 
1 c + d cd 
1 c + d cd ; 
the factors s — c,t — c,u — c can next be removed and the remaining co- 
factor left in the form 
1 d . . . 
1 d 
1 d 
1 d 
1 (vivxy)-^ ( vwxy) 2 ( vwxy) z ( vwxy ) 4 , 
which is equal to 
{v — d){w -d)(x— d)(y -d). # 
(6) We are now in a position to show in a variety of ways that © does 
not vanish identically. Probably the most interesting way is to be 
found in the evaluation of it for the case where 
g,l,i = d , c , e , 
for then it takes the form of the product of eighteen differences. To 
see this we have only to note that its (3, l) th element then vanishes from 
having two rows alike, that the (3, 2) th element vanishes for the same 
reason, and that the (2, 2) th element vanishes because it has three rows 
with two variables in common. © thus reduces to one product of three 
elements, namely, to 
