VOL. XVI.] PHILOSOPHICAL TRANSACTIONS. 877 



tion, or towards the contrary point, if it be + /»• Moreover, from the point 

 D, or from C if the quantity p be not in the equation, D E = -J- 9 is to be 

 erected perpendicular to the axis, to the right hand if it be — y, but on the 

 other side of the axis if it be + q ; and then a circle described on the centre E 

 with the radius A E, if it be only a cubic equation, will intersect the parabola in 

 as many points F and G, as the equation has true roots ; of which, the affirma- 

 tive ones, as G K, will be on the right side of the axis, and the negative ones, 

 as F L, on the left. But if the equation be a biquadratic, the radius of the 

 circle should be either augmented or diminished, by adding if it be — r, or sub- 

 tracting if + r, from its square, the rectangle ar, which rectangle is the pro- 

 duct of the latus rectum and the given quantity r, which is very easily done 

 geometrically. And, letting fall from them perpendiculars to the axis, the in- 

 tersections of this circle with the parabola will give all the true roots of the 

 biquadratic equation, the affirmative on the right side of the axis, and the 

 negative on the left. Here it is to be observed, that I endeavour to have the 

 affirmative roots on the right side of the axis, to avoid the confusion necessarily 

 arising from a great number of cautions, where the reason of them is not 

 evident. 



Having premised these things, in order to make way for the construction of 

 these equations, where the second term is found, we are to consider the rule 

 for taking away that term, and .reducing the equation to another, such as might 

 be constructed by the preceding method. Now all cubic equations of this class, 

 are reducible to this form, «^ hzz. apz. aaq = O, or to this, z'. bzz. sic. aaq 

 = O; biquadratic equations to this form, z*. bz^. ajbz*. aaqz. a^ r = o, or to 

 this, z*. bz^. >)c. aaqz, a' r == O, or 2*. i«'. aj&«*. >|<, a' r = O, or lastly to this 

 form-, 2*. bxii i^. >|<. a' r = O; from all which there arises a great variety, ac- 

 cording as the signs + and — are differently connected, and hence the general 

 rule serving for all these cases is rendered very obscure and difficult, unless it be 

 cleared up by the following method, and freed from its intricacies. 



The second term in biquadratic equations is taken away by putting x = % -\, 

 ^ b \{ It he -\- b m the equation, or * =« — -j- 6 if it be — b; hence x — -J- A in 

 the first case, and ar -|- 4- A in the second, is = « ; and instead of z, in any pro- 

 posed equation, substituting its equal, there will arise a new equation, wanting 

 the second term, of which all the roots x are either greater or less than thfe 

 required roots z, by the given difference 4- b. 



Example 1 . If z* -\- bz^ — apzz — aaqz -)- a' r = O. 

 Put ^7 — 4- i = z. Then it will be, 

 x^-^ix+^b' = z% 



<C^- ^bx^ + ^b^X~-^b^=: Z% 



x*-bx' + lb-x'-^b'x + ^i^b* = z\ 



VOL. III. 3 C 



