408 Proceedings of Royal Society of Edinburgh. [sess. 
a formula that from its very nature suggests and proves a wide 
extension of itself.” (xxm. 10) 
/ 
It belongs evidently to the class of vanishing aggregates of pro- 
ducts of pairs of determinants, of which so many instances have 
presented themselves. There is a manifest misprint in the third 
product, which should surely be 
£PD (cfg . . . &)x£PD (ab . . . 1); 
and there is an error in the signs connecting the products, which, 
instead of being all + , should be + and - alternately. When the 
determinants involved are of the third order, the theorem in the 
later notation is 
K/a&MViAl - \ h ifi9z\-\ a i c i d %\ + \ c ifiS sl-KMsI - l^iMl-KVsH 0 . 
which is readily recognised as an identity given by Bezout. 
With this theorem the paper proper ends, but in a postscript an 
additional theorem of a curious character is given. As enun- 
ciated by the author — even his double mark of exclamation being 
reprinted — it is (p. 43) : — 
“ Let there be (n - 1) bases a,b,c,. . . , l , and let the argu- 
ments of each be “ recurrents of the n th order,” that is to say, 
let 
Let It* denote that any symmetrical function of the r th degree is 
to be taken of the quantities in a parenthesis which come after 
it, and let ^ indicate any function whatever. Then the zeta-ic 
product, 
£(£R ,(abc ... 0 x i^PD(0abc . . . 1)) 
is equal to the product of the number 
