Of CUBIC EQUATIONS. 115 



and fubftituting —- for r, and —- for z, we derive thefe two 



equations, 



p — ypv 2 — ^ 3 

 q =z m 3 — - 3^ 



whence it is manifefl that p is a divifor of />, and v a divifor of 

 <7, as before. It will be eafier for trial to write the equations 

 thus : 



g-ft 



rv z — a 



And let it be obferved, that we may here give to jju and v any 

 figns conliftent with the condition, that p ~*~ f * and rv ~~ q (the 



values of v x and i o< 2 ) are pofitive numbers. 



It is to be remarked too, that, in this cafe, z has three values. 

 If, however, we can find one value this way, the two others are 

 readily obtained. For if v be one value of z, the two other va- 

 lues are *""*"*' \ - and V T , : becaufe thefe values are the 

 1 — v v 3 1 -+■ i> v 3 



tangents of two arches that differ from the arch of which v is 

 the tangent by 120 and 240 °. 



Though this is a matter more curious than ufeful, I mall 

 add one more example for the fake of illuftration. 

 Let the equation be 



x* — 39X 2 + 479.V — 1 88 1 =0. 

 Here A = — 39 ; B =. + 479 ; C = — 188 1 : And 



M = + 404 

 N =: — 4724 

 m = -f- 16976 

 n = + 1584 

 Qj= — 2560Q 

 And the equation belongs to Cafe L 



a 



