When stylus A follows curve y=f(x), carriage 1 (Fig. a) moves along axis x-x, and sliders 2, 3 and 4 move along the guiding members of carriage 1 in the direction of the y-axis. Sliders 2, 3 and 4 are connected together by lazy tongs linkages which operate as adding mechanisms. Besides, the middle points N and B of these linkages are connected together by a third lazy tongs linkage for which the following relationship holds true F͞D’=(1/4)(E͞E’+2D͞D’+K͞K’). At points M and H (Fig. b), sliders 3 and 4 are connected by turning pairs to sharp-edged wheels 5 and 6 whose axes move, by means of parallel-crank linkages, perpendicular to guiding levers 7 and 8. Stylus P describes a curve whose ordinates are determined by the equation D͞D’=∫(K͞K’/0₁͞K’)dx=(b/K₁)∫(dz/dx)dx=(b/K₁)z+K where K₁=0₁͞K' is a constant. Stylus C describes a curve with the ordinates K͞K’=∫(F͞D’/0₂͞D’)dx. Taking the integration constant to be K=0, and since 0₂͞D'=K₂ is a constant, we have F͞D’=(1/4)[f(x)+2(b/K₁)z+b(dz/dx)], K͞K’=∫(F͞D’/K₂)dx=∫(1/(4K₂))[f(x)+2(b/K₁)z+b(dz/dx)]dx=b(dz/dx), 4bK₂(d²z/dx²)-b(dz/dx)-2(b/K₁)z-f(x)=0 If at x=0 styluses P and C are set to the given initial values bz/K₁ and b(dz/dx), then when stylus A is moved along curve f(x), styluses P and C trace in a definite scale the integral curve with the equation az"+bz'+cz+d=0 and the curve of its first derivative.
$1302$LG,MO$