An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

INVERSIONS AND DETERMINANTS

versions among the letters, there being no inversions among
suffixes.
28. Let the determinant be represented by Ia b.....n, r.
Then the scheme of development with respect to natural order
of rows will be identically the same, with regard to symbolic constituents, as it was in the development of 1 al b2.... mnlc with
respect to natural order of columns; but only the symbols of
the principal diagonal represent the same constituents of the
determinant in the two cases. Either scheme, however, must
lead to the development of the determinant represented since
both accord with the definition of a determinant.
Examples.
2-6 4
Develop 5 -3 9. Represent it by al b2 C3 Ic.
-7 -8
Then letters correspond to columns and suffixes to rows.
Hence selecting from columns in natural order and permutingt
the orders of choice from rows is indicated by writing alb2cs,
alb-c2, a2blcs, a2b3cI, a3blc2, asb2cl; the letters being written in
natural order and the suffixes permuted in all possible orders,
each permutation corresponding to a term of A. Counting the
inversions among suffixes in each term, attaching corresponding signs and substituting the values of the constituents, we
obtain the 3! terms of A; thus
Z (-I)i (albAc3) - +.(2)( -3)(-8)-(2)(I)(9)-(5)( -6)(-8)
+ (5)(I) (4)+(-7)(-6)(9)-(-7)(-3)(4)
- 48-18-240+20+-378-84
104
Scholium. To follow the symbolic definition (Art. 2I) to the
letter we should write
A = (-I) (alb2c) + ( —)1 (alb3c2) + (-I)1 (a2blcs) + (-I)2
(a2b;cl)) -- (-i)2 (ablc2) +- (-I)3 (a3b2cl); but this is unnecessary as the number of inversions in any term is +, or —, according as the number of inversions in that term is even or odd.
2. Represent A i:l Ex. I. by I alb-c:3 r, and develop. Compare results. Compare Art. 28.
3. Let the order of choice from columns in Ex. I. be b c a
for every term. Compare results.
Suggestion. i E number of inversions among both letters
and suffixes.
I 2 I 3 i
Develop AnlS. 55.
2414.