378 PHILOSOPHICAL TRANSACTIONS. [aNNO 1687 . 



Hence x* — bx'+ ^b^x^ — ^b^x + ^-^ b* = + z*, 

 + ba^—-i-b^x' + -^b'x—^b* = + iz', 

 — apx^-\- -^apbx — ^apb^= — apz", 



— a* ^ar + 4- ^^ 9^ = — a^qz, 

 + a^ r = + aV. 



The sum of all these is a new equation, wanting the second term, and which 

 consequently may be constructed by Descartes's rule, by taking, instead of-^p, 

 half the co-efficient of the 3d term divided by the parameter a, that is — 



-^ -Lp; and instead of ^q, half the co-efficient of the 3d term divided by 



a^, that is -| — ^— +7 4-0. The members of which that have the sign 



+, are to be set off from the axis towards the left hand, and those that have 

 the sign minus (— ) to the right, in order to find the centre of the circle re- 

 quisite for the construction, and of which the intersections with the parabola, 

 drawing perpendiculars to the axis, may denote all the true roots x, viz. the 

 affirmative on the right hand side of the axis, and the negative on the left. But 

 since x — ■\-b= z, by drawing a line parallel to the axis, on the right hand 

 side of it, and at the distance of -J- b, the perpendiculars terminated by this 

 parallel, will denote all the roots required z ; viz. the affirmative ones on the 

 right, and the negative on the left. 



As to the radius of the circle, it is had, by adding the negative, and deduct- 

 ing the affirmative parts of the 5th term divided by o*, from the square of the 

 line A. E, drawn from the centre E, to the vertex A of the parabola : which is 

 best done, by taking, instead of A E, the line EO, terminated at O, the in- 

 tersection of the parabola and the above-mentioned parallel ; for its square com- 

 prehends all the parts of the 5th term brought into the new equation by the 

 taking away of the 2d term, as is easily proved. And there only remains to in- 

 crease the square of EO, if there be — r in the equation, or to diminish it if 

 it be -|- r, by the addition or subtraction of the rectangle a r, from whence the 

 square of the radius of the circle sought is composed. 



This is the method of investigating Baker's central rule, which is sufficiently 

 easy and free from all cautions ; and the only difference arises from hence, that 

 I determine the centre of the circle by the axis, and Mr. Baker by a parallel to 

 the axis ; and that I always find the affirmative roots on the right side of the 

 axis, which he has sometimes on one side and sometimes on the other. 



As to cubic equations, they are to be reduced to biquadratics, before they can 

 be constructed by the same general rule ; which is done by multiplying the pro- 

 posed equation by its root z, whence arises a biquadratic equation, wanting the 



