260 Mr. H. Cunynghame on a Mechanical Method 



represent an error which looks much larger in the pressure. 

 In comparing the errors of calculation from other empirical 

 formulae with those due to Rankine's formula, it is well to 

 remember that Rankine calculated pressure, and that his 

 errors, which for that reason might have been expected to 

 appear large, were very small, thus showing the great accuracy 

 of his formula. 



XXXVI. On a Mechanical Method of Solving Quadratic and 

 Cubic Equations, whether the .Roots be real or impossible. 

 By Henky Cunynghame, Barrister-at-Law* . 



npHE method which I have the honour to bring before this 

 J- Meeting depends upon the use of a parabola of the form 

 represented by the equation x n =y. And it is capable of 

 effecting solutions of all equations of the form M n -\-mx = C. 



Let us first examine the case of a cubic equation. By 

 Cardan's rule reduce the equation to the form a? + A# + B = 0. 

 In fig. 1 let P Q be a cubical parabola, such that the ordi- 



nate at any point represents the cube root of the abscissa 

 along the axis of X, measured negatively towards the right. 

 That is to say, let OM= (PM) 3 . From P draw any line such 

 that the angle PNM = cot" 1 A, and let ON = B. Then it is 

 clear that 



OM + MN + ON = 0; 

 that is, 



(PM) 3 + PM cot PXM + ON = ; 



or, putting PM=^, cotPNM = A and OX = B, we have 



x z + kx + R = 0. 



And this, therefore, shows that if from a point N, such that 

 OX = B, we draw a line XP inclined at such an angle with 



* Communicated by the Physical Society : read December 12, 1885. 



