TRACT 28.

AND INFINITE SERIES.

S7

31. From the bare inspection of this table, several usefuland curious observations may be made. And first it ap-pears, that when q is positive, as in all the forms after the12th, r is the greatest root; but when q is negative, or inall the cases to the 12th, r is one of the less roots.

32. In all cases before the 4th form, r is the least root,

because or —~, &c, is always greater than 1.;

and in all such forms, (iqY is less than (fp) 3 ', but the for-mer approaches nearer and nearer to an equality with thelatter, till the 4th form, where (iqY is become zi (tPY>and r is then equal to one of the other roots, because«/9 —1 _ ? _ 1

2 2

33. From hence r becomes the middle root, and con-tinues so to the 12th form, where it becomes equal to whathas hitherto been the greatest root, and the other root be-comes at this place = 0 ; and (jq)' has decreased from the4th form, all the way more and more, in respect of (a/;) 3 ,till at this 12lh form it has become = 0, or infinitely lessthan (jp) 3 -

34. From this place, r becomes the, greatest root, thesign of q changes to +, and (jq) 3 again increases in respectof (jp) 3 , till at the 13th case it becomes again equal to it,and the two less roots equal to each other, like as at the 4thform.

35. From hence (jq) 2 becomes greater than (-*• p) 3 , and in-creases more and more in respect of it, till at the 16th step,where p is = 0, or (jq) 2 infinitely greater than (jp) 3 -

36. From this place the sign of p becomes +, and (jq) 2continually decreases in respect of (fp) 3 , to infinity,

37. By help of this table, we may find the roots of any

cubic equation x l px - q, whenever we can assign the/relation between ^/p and ^q. For since one root r is al-ways = = = l /—and the other two roots

J (nzpl)p v fct:3 v ?/^;r

«= — Ff(1 + ±.n), it folloivs, that if, from the equation