156 Journal of Agricultural Research voi.xii.No.4 



where A = DG. (27a) 



The direction cosines X, m, v, of the angles which the new X-axis makes 

 with the original axes are obtained from the following: 



(a—ki)\+hij.+gv = (28) 



h-\ + {b-k,)ti+jv = (29) 



9X+/M + (c-fei> = (30) 



2X+m2+v2=i. (31) 



Similarly, the direction cosines of the angles which the Y- and Z-axe 

 make, after rotation, with the original axes are found by substituting 

 k^ and ^3, respectively, for k^ in equations (28), (29), (30), and (31). 



When equation (27) was developed from the surface of a plow bottom 

 having two sets of straight-line generators, it had the following general 

 form: 



This is the equation of an hyperboloid of one sheet, a vase-shaped figure, 

 the skeleton of a section of which is shown in figure 3. When 2 = 



equation (32) becomes -^ + r2=i, and the cross section through the 



plane z=0 (fig. 4) is an elUpse. When y = 0,\.\\e equation becomes 



-^ — -3= I, and the section through the plane y = (fig. 5) is a hyperbola. 



Similarly, when % = 0, ^--3= i (fig. 6). Figure 7 indicates the two sets 



of straight-line generators which lie on the surface of an hyperboloid of 

 one sheet .-^ 



APPLICATION OF THB DEVELOPMENT TO A PROBLEM 



In order to develop the equation which will describe the surface of a 

 plow bottom, it is necessary to obtain the data called for in equations 

 (16), (17), (18), and (19). This application of the development will be 

 carried through for the bottom represented in Plate 6, A, which bottom 

 was placed upon the machine shown in Plate 7, B, so that the origin of 



JThe constants a, b, and c of this equation do not necessarily have the same numerical values as in 

 previous equations. 



* The method for obtaining the equations of any line on the surface is given in Snyder, Virgil, and 

 SiSAM, C. H. op. cit., p. 93. 



