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

8

same as the number in the written line. The corollary follows
from this and Theorem III.
8. DEFINITION. The order of a group, of two or more complex symbols, is the sum of the orders of the constituents of
the group.
9. Theorem IV. The number of inversions in any line of
doubly complex symbols, due to the presence of anly specified two or
more such symbols, is even or odd according as ( I) + i is ~ or -
s being the order of the specified group and i its number of iversions.
Let S be the sum of the numbers of inversions respectively
due to each symbol of the specified group considered separately. Then S includes each inversion occuring among the
specified symbols twice, and each inversion between a symbol
within the group and another without the group b_:t once.
Therefore S —is the number of inversions due to the specified
group in the line. But S and s are both even or both odd
(Theorem III and properties of numbers). Therefore the
number of inversions due to the group is even or odd according as S-i, and consequently as s-i, is even or odd; that is
according as (-I)s-i, and therefore as ( —) s+i, is - or -.
Cor. Tzheorem III is a special case of Theorem IF, since thzen
s r and i = O.
Examples.
i. Is the number of inversions in a2 clf7 g, d3 e4 b,, due to
the presence of a2 g6 d b,, even or odd? For ease of enumeration represent the groups byI 3 6 7 4 5 2 and I 7 4 2
2 I 7 6 3 4 5 2 6 3 5'
In the latter (s —i)30-5=25, and (-I)25 is —. Hence there
should be an odd number of inversions due to I — 4 2; and by
263 5
actual count there are 17 in I3 6 7 4 52 and 2 in 3 6 5, leav2176345 I74
ing 15, an o0(t number, due to I 7 42.
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2. In Ex. I S-i=24 for 3 6 5 and accordingly there are
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12 inversions due to 3 6 5
I 7 4
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3. In d( a2 c4 bu e3 there are 6 inversions due to the presence
of c4 c 6'e 1 for which s ---i' i6 an evencz number.