396 Proceedings of Royal Society of Edinburgh. [sess. 
done tons par a a , et puisque les signes de ces termes sont de- 
termines d’apres la maniere exposee plus haut, on tronvera 
leur somme — a a (bc . . . r, /3y . . . p).” 
Vandermonde’s theorem regarding the effect, on the function, of 
interchanging two bases is stated generally, and a demonstration is 
given. The mode of demonstration, which occupies one page and a 
half, will he readily understood by seeing it applied in later nota- 
tion to the case where there are four bases, that is to say, where the 
theorem to be proved is 
I a a bpc y d s | = - | b a apc y d 8 | . 
By repeated use of the recurrent law of formation we have 
| ttabpCyds | = a a | bpc y d 8 | - ap \ b a c y d s | + a y \b a Cpd s \ - a 8 1 b a c^d y \ 
= a a { bp | c y d s | - by | Cpd 8 1 + b s \cpd y \} 
~ ap{b a \Cyd s \ - by | Cads | + b 8 \ c a d y |} 
+ a y {b a \ cpd s | - bp\c a d s \ + h\c a dp\) 
- a s {b a | cpdy I - bp | c^y \ + b y \ c a dp | } . 
By collecting the terms which have b a for a common factor, bp for 
a common factor, and so on, this result becomes 
| O/gbpCydfr | ~ b a ^ap | Cyd$ j cjy| Cpd§ | cpdy j j - 
+ bp{a a | c y d s | - a y \ I + «a| c od y 1} 
-by\a a \ cpd s \ - ap\c a d s \ + a s \c a dp\} 
+ bs{a a \cpd y \ - ap\c a d y \ + a y \ c a dp |} , 
= - b a | apCyds | + bp\a a b y ds\ - b y \ a a cpds | + &s| a a cpd y | , 
— [ bofipCyds | , 
as was to be proved. (xi. 5) 
The suggestion readily arises that this -process would be equally 
applicable in proving Vandermonde’s theorem regarding the vanish- 
ing of a function in which two bases are identical, and the process, 
it may be remembered, was actually so employed by Desnanot. 
One of the theorems given by Scherk, and later by Drinkwater, 
appears in the following form (p. 240), the peculiar notation adopted 
for a determinant with a row of unit elements being constantly 
employed throughout the remainder of the essay: — 
