214 Investigation of Determinants. (April, 
represent the corresponding values of the preceding 
quantities when the Jeading constituents are all zero. 
Also let [E,] be the corresponding value of E,, 
And let “C, be the number of combinations of n things 
taken 7 together, so that— 
sd 5 — 
| 1 n 
="Cy_re 
| 7. |n-r 
It is shown (Salmon, Art. 40) that in general— 
dA d2A 
= (A) + 24 aypeLaas |} + 34 aspen Exe = re 
+34 AppUaqhyrrs crac mee 
+ ose wt eo wt Gy-0ete © > Gan) eee 
Hence, noting that the number of ways in which a con- © 
tinued product, as (a,;.42,43; . . + » Ar) of 7 constituents 
can be formed from the 7 leading constituents (of type ap») 
is “C,, also that the number of elements in an vth minor of 
type— 
dra ] 
day1.ddz2.da33 ..- + day)’ 
(being a determinant of (~—v) rows whose leading con- 
stituents vanish), is by above notation [E,_,], also that by 
the notation the number of elements in A is E,, and in [A] 
is [E,], it results that— 
En= [Ex] +°C,. [En-1] +%C,. [En-2] +... + . +°%C,. [En—o] 
vere -2 [E, ] +7Cy-x LA ] +1. 
= [Ey] +2. [En] feces) [Epaal +. to 
| 7. | —9 Yr 
, +t) TE] An [E:] +1... (4). 
Now the numerical coefficients are the same as in the 
expansion of the binomial (x+1)”, and the suffixes of [E] 
are the same as the indices of x in that expansion; hence 
modifying the notation with the interpretation— 
[E]?= [E;], (s) 
[E]*. (E] = [E]*#4= [Eps] 
that is to say, making the suffixes of [E] follow the index 
law, Equation (4) takes the following remarkably simple 
symbolic form :— 
B,=((E] 3)". wove a ee (6). 
Further, this Equation being general for all (positive integral) 
values of n— 
er cabelas 
