Jan. 28, 1918 



Study of Plow Bottoms 



179 



generated by laying a straightedge upon oe and moving it backward 

 parallel to the plane z=0 with the lines e^h and em as directrices. 

 This surface will be a portion of a hyperbolic paraboloid, the same 

 general type as the surface which Jefferson proposed. The orthogonal 

 projection of the generator in various positions upon the plane z=0 

 will look as shown in figure 23. For stiff, clay soils the lines (fig. 24) 



e o 



From Report of N. Y. State 



Agric. Soc. 1867 



Fig. 23. 



e O 



From Report of N. V. State 



Agric. Soc. 1867 



Fig. 24 



e O 



From Report of N. Y. State 



Agric. Soc. 1867 



Fig. 25. 



are made concave and for loose, sandy soils (fig. 25) they are made 

 convex. As no exact description was given regarding the shape of 

 the curves (fig. 24, 25), it has not been possible to develop equations 

 for the surfaces. However, as it is known that these surfaces have 

 straight lines in one direction and can not be described by an equation 

 of the second order, they are of the fourth order or higher. 



KNOX'S PLOW BOTTOM* 



In 1852 Samuel A. Knox, of Worcester, Mass., applied for a patent upon 

 the surface of a plow bottom which was certainly unique. The skeleton 



of this surface is shown 

 in figure 26. The seg- 

 ments of circles I, 

 II, and III are placed 

 in parallel planes 12 

 inches apart, so that a 

 series of straight lines 

 will cut the three cir- 

 cles. Circles I and III 

 have equal diameters 

 and the diameter of 

 circle II is one-half 

 that of circles I and 

 III. As the equation of this surface is of the eighth order, it will not be 

 worked out in detail, but a development will be given to show how the 

 equation could be obtained. 



Let the equation of the three circles be ^ 



from ifepon of r*Y stole Agric Soc /e67 



Fig. 26. 



Z=0, 



{x-aY+iy 



z=k 



H'^' 



' Gouij>, J. S., et al. Op. cit., p. 49-. 



* This development is the work of Virgil Snyder, Professor of Mathematics, Cornell University. 



