and Infinite Series. 401 
35. From hence becomes greater than J/^ 3 , and 
increafes more and more in refpeit of it, till at the 1 6th 
ftep where/' is = o, or infinitely greater than I/ 13 . 
36. From this place the fign of p becomes + , and 
\ 3 
±q' continually decreafes in refpe<5t oi"-p to infinity. 
37. By help of this table we may find the roots of 
any cubic equation x l + px = q whenever we can affign 
the relation between Vp and J/q. For fince one root r is 
always = ” —hi — ./-if- = */— if-, and the other two 
roots = — \r . 1 ± V d=n t it follows, that if from the 
equation sj ~ ^ = ^7=77* where the two denominators 
under the radicals differ by 4, we can affign the value of 
n, the above formula will give us the roots. 
38. As if the equation be x z - 18a: = -27. Here 
p = 18, and q = 27 ; then y/if -\l\ =^9 = 3, and 
,V 4? = 4/27 = 3 alfo; therefore n + 3 = 8, or n - 1=4, 
V 4 
either of which gives n- 5 : confeqnentlv, r = p~~j, 
_ 8? _ 2j _ _ - s t j ie middle r oot, becaufe — is found 
4 P P * 4 P 
between the 4th and 12th cafes, which are the limits of 
the middle roots: and — \r.x±Vn — — \ . 1 \/ 5 = 
4*854102 and 1*854102 are the greateft and leaft 
roots. Or, thefe two roots may be alfo found in the fame 
manner 
