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.

2S INVERSIONS AND DETERMINANTS
46. Homogeneous Equations. In the case of (n —I) homogeneous equations containing n unknowns we cannot determine the unknowns, but their ratios may be expressed in the
manner of the following:
Let al x+-b y+cl z o,
a2 x+b2y+-c z 0o,
be two homogeneous equations in three unknowns. Add below another equation of the same general form,
a3 x+b3y+c3 z = X,
a3, b, e3, x, being undetermined, and represent the array of coefficients by A.
o bl Cl
o b2 C2
X b3 c3 X A/ x g x Cf
x /A- - 'A A z
x y z
3 = C3
The method is easily extended to any number of equations.
Example.
2 w+3 x- 4Y+15 z = o,
4 w- X+12Y-2 Z = 0,
6 w-7 x —20y+ = zo...,/1 - 376, B = - 752, C4 = 188, 2D4 = 752.
w x ' z
2 4 -I -4