244 On a Machine for Solving Equations. 



a increases algebraically, it approaches the vertex, and the 

 two roots become more nearly equal until at the vertex they 

 join, and there are two equal roots; here of course 6 2 = 4a. 

 If the sliding line is below this position, it misses the parabola, 

 and there are no real roots. 



Now going back to the case of real roots, it is clear that 



they are equally greater and less than +r, which is the 



distance of the vertex of the parabola from the axis of y, and 

 the excess or defect is the half width of the parabola at the 

 place of cutting. Now this half width is by the nature of 

 the parabola equal to the square root of the distance along 

 the axis from the vertex to the sliding line. If, then, the 

 sliding line misses the parabola, the half width of the curve 

 at the position of the line being equal to the square root of 

 its distance above the vertex, will, as this distance is now 

 negative, be impossible, or the curve at this region does not 

 exist ; nevertheless the imaginary half width is still the square 

 root of the distance of the sliding line above the vertex. 

 Geometry is incapable of representing this impossibility ; 

 but it is curious that the machine can be made to indicate 

 its existence and measure its amount. Since the ordinates 

 of the parabola represent the moments on the first beam for 

 every value of «#, it is clear that a minimum of moment will 

 be produced in the case of impossible roots. If, then, the 

 machine fails to find two real roots (which of course must 

 be within its limits), the first beam must be constrained by a 

 spring instead of being free. The deflection will then indicate 

 moments, and therefore the position of the minimum may be 

 observed. The reading on the scale of the instrument is then 

 the real part of the root, and the square root of the moment, 

 which may be found by applying weights, the impossible 

 part. 



In the case of a cubic equation there must be one real 

 root, and there may be three ; if only one exists this may be 

 found and divided out, when the resulting impossible qua- 

 dratic can be solved by the machine. If all these roots are 

 real, they can of course be found directly. 



In a biquadratic, or one of a higher order, the possible 

 roots may be found. If there remain more than two im- 

 possible roots, the machine is incapable of finding them. 



While attempting to find properties of such equations that 

 might be made use of in the construction of a machine, I 

 found a curious relationship between two pairs of curves 

 which might be turned to account in a curve-tracing machine. 

 Thev are probably well known, but if so it may be worth 



