Of CUBIC EQUATIONS. 115 



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

 equations, 



q = ru^ — 3^^, 

 whence it is manifeft that ,0, is a divifor of p, and i/ a divifor of 

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

 thus : 



m 3''/* 



2 _ rv'' — q 



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

 figns confiftent with the condition, that ^7"^ ^^^ " ~ '^ (the 



values of v* and |M.^) are pofitive numbers. 



It is to be remarked too, that, in this cafe, 2 has thi'ee 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 ""T ^ : becaufe thefe values are the 



I — '2'v 3 I -+• 'Z'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 tifeful, I fliall 

 add one more example for the fake of illuftration. 

 Let the equation be 



x^ — 39^^ + 479* — 1881 =0. 

 Here A = — 39; B = + 479 j C =: — 1881 : And 



M zi + 404 

 N = — 4724 

 m = -{- i6gj6 

 n =z + 1584 

 Qj:: — 25600 

 And the equation belongs to Cafe I. 



