284 Mr. Wilson's Essay on the 
is merely employed to obtain the difference of the roots ; that 
(upon analysing the formula) the part under the vinculum is 
always that difference, and nothing else, and why it must 
be so. 
19. Next, let (71 = 3), and the equation be the complete 
cubic ( x 3 — px 1 -j- qx — r = o). If we make it our first step 
here, as in the last case, to inquire what is necessary to con- 
struct a general representation of three numbers in simple 
equations, we shall find it must consist of the same parts, the 
sum and the differences : but, as the differences increase in 
number, to show the order in which they are taken, and the 
law they observe progressively, I shall subjoin a general table 
of the simple representation of the different orders of quantities. 
As in every equation the sum of the roots is always given, I 
shall, for greater simplicity in my table, suppose it always to 
vanish. If then there be a series of general equations, begin- 
ning with a quadratic, and proceeding upwards with progres- 
sive indexes, in all of which the coefficient of the second term 
(p) be taken = (o), and (A) be supposed a difference of the 
roots of the first, (A) and (B) two of the differences cf the roots 
of the second, (A, B, C,) three differences of those of the third, 
and so on ; in taking of which differences, no other caution is 
necessary than that they should be similarly situated, viz. all 
derived by comparing the same individual root with the remain- 
ing ones, as if (a) be taken as a root, and (a — 6) be the first 
difference, ( a — c, a — d, a — e &c a — n), having all the 
same antecedent letter (whose number will always be in — 1)), 
must be the rest ; then , the table will be as follows : 
