of Solving Quadratic and Cubic Equations. 261 



OX that tanPNM = A, then the value PM will represent a 

 root of the equation ^ 3 + A^ + B = 0. Of course, if PN pro- 

 duced cuts the other branch of the cubical parabola, there 

 will be two other real roots. If it does not, those other roots 

 v/ill be impossible. 



In making a machine practically, it is well for convenience 

 to draw on a paper a cubical parabola, say 2 feet wide, 

 and make 1000 divisions on OX each way. Then make 

 XH= v / 1000 = 10. But as this would make the parabola too 

 thin, lot us multiply its ordinates by 10, and then take a co- 

 tangent protractor scale with the cotangents also multiplied 

 by 10. So that the parabola drawn, as well as the tangent 

 scale in any position, is an orthographic projection of the real 

 one. Such a machine will find real roots true to two places 

 of decimals. 



To find the impossible roots. First find the real root, say 

 P ; draw P Q to touch the other branch of the parabola in Q. 

 It may then be easily shown (though I am not aware that it 

 has been before pointed out) that 



and that therefore 



PM 3 

 o 

 And, further, 



SAY SQ _ 1 

 HW PM~2' 

 and 



SW + MW = 80 + OM=^+PM 3 =f PM 3 ; 



3SW=|PM J 



whence 



and 



SW=|PM 3 , 

 and 



MW=-|PM 3 . 



Now if a be the real root of the equation, so that PM = « : 

 then, by dividing (x — a) into ^ ,3 + A^ + B = 0, we can easily 

 show that the other two imaginary (or real) roots are given 

 by the expressions 



-|± v-j* 2 +A, 



B being =« 3 + A«. But we have already shown that 

 }a 3 = f PM 3 = MW, 

 Phil. Mag. S. 5. Vol. 21. No. 130. March 1886. T 



