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.

30

INVERSIONS AND DETERMINANTS

or resultant of the system; the reader is referred, however, to
the higher theory of equations for the precise definition of the
reszltazt.'it
Examples.
Eliminate a and y from the three equations
al X+bl y+-c z+-d = o,
a2 x+-b2a Y+C2 Z+ — = 0,
a3 x+b3Y+-c3 z+d3 =: o.
al bl clz+di
Ans. a2 b2 c2+-d2 = o.
a3 bs C2,+d3
48. Sylvester's method of eliminating x from any two
rational, integral equations in x is illustrated by the following
examples:
i. Eliminate x from the equations
ax2+bx-c = o and ax2+-x+y = o.
We derive ax3+bx2+cx =o,
ax2+bx -j-c =o,
ax3+~f 2+yx =o,
ax2+-I+Y -0.
Whence, treating x3, X2 and x as so many unknown quantities we obtain
a b c o
o a b c |
abco
oabc
= o;
an f o
an equation free of x.
2. ax:3+bx-+cx+d = o and -t2+qx+;r - o.
a b c (1 o
o a b c ed
Ans. f q r oo o.
op j q r o
o o p q r
Scholium. It should be observed that the two given equations cannot be simultaneous unless the coefficients are sucli as
to cause the eliminant to vanish.

* See Burnside and Panton on Theory of Equations, Vol. II, p. 69.