for Solving Equations. 24* 



work possible there is a practical limit beyond which it is 

 hardly possible to go. 



The only difficulty lay in the design of the beams, which 

 would allow the connections between them to be capable of con- 

 tinuous variations at all parts of their length. The joint must 

 be able to slide freely past the axis and past the coefficient 

 pans. All the beams are of the same construction except that 

 alternate ones have the connecting joint on opposite sides of 

 the centre, as shown by the diagram fig. 1. Fig 2 is a front 

 view of a beam with its supporting-rod and coefficient hooks 

 complete ; fig. 3 a back view, and fig. 4 a plan of the same. 

 Fig. 5 is a transverse section through the middle, and fig. 6 

 an end view of a beam and connecting joint but without sup- 

 porting bar. It will be seen that there runs along the back 

 of each beam a web which may be embraced at any part by 

 a pair of fingers fixed to the front of the next beam behind it, 

 i. e, by the beam which is next higher in order of number. 



It is evident that this construction allows the connecting 

 joint of each beam to slide past the centre of the next lower 

 one. 



The supporting bars hang from a series of longitudinal 

 beams lying in a cradle, the alternate ones being joined 

 together. These then can be shifted relatively past one 

 another, and one set can carry an index reading on a scale 

 on the other set, so placed as to read zero when the distance 

 of the centres of consecutive beams is one unit. 



To show how to apply the machine to find the impossible 

 roots of a quadratic, it will be necessary to consider the qua- 

 dratic equation y = a -+- bx + ex 2 as representing a parabola. In 

 this parabola a has nothing to do with the shape of the 

 curve. Variation in a only shifts it or makes it slide up 

 and down the axis of y. It may therefore be struck out for 

 the present. Then the parabola passes of necessity through 

 the intersection of the axis of m and y. If the sign of b is 

 changed, the new parabola will be a reflection of the old 

 about the axis of y. Fig. 7 thus represents the double equa- 

 tion x 2 ±bx=y. 



The distances of the vertices of these parabolas from the 



axis of y are + ~ , and the values of y at the vertices are each 



b 2 . 



— -p Now let a line parallel to the axis of x slide down the 



axis of y, and let its distance above the axis of x be a in a 

 series of equations a + bx + (c)x 2 =y (the c may be omitted), 

 then those values of x at which this line cuts the parabola 

 will be roots of the equations. As it slides down, that is as 



