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.

CHAPTER VII
ELIMINANTS AND DISCRIMINANTS
47. Elimination. To eliminate the two quantities x, y, from
the three equations
ai x+bi y+ci = o, (I)
a2 x+b yc2 = C o, (2)
a3 x-b3 y+C3. (3)
Multiply (I), (2), (3), in order by C1, C2, C3, and add The
a1 bl c1
sum is cl Cl+c2 C2+c8 C3 = a2 b2 C2 = o.
as7 b3 C3
This expresses the condition that the three equations are
simultaneous in x, y; that is consistent, or capable of being satisfied by the same set of values of x, j.
Similarly thefour equations
<a x+bly-+-cl z+d =- o,
a2 x+b2 y+c2 z+d2= o,
a4 x+b4y+c4 z-Kd4 - o,
between the three quantities x, y, z, are consistent if
al b\ ci di
a2 b2 C2 d2,3 ba C3 d3
n4 b4 C4 d4
In general n-I quantities involved in a system of ti simultaneous equations may be eliminated by arranging the terms
of each equation in corresponding order in one member and
equating to zero the determinant of the array so formed by
the coefficients. This determinant may be called the eliminant