152 



Journal of Agricultural Research 



Vol. XII, No. 4 



backward and forward upon guides so marked that the board in all posi- 

 tions will be squarely across the guides. When a plow bottom is properly 

 placed upon the platform the x, y, and z coordinates of any point upon 



the surface can thus be recorded upon coor- 

 dinate paper fastened upon the drafting 

 board. 



/ 

 / 



/■>^ey6'26y 





a'^zVs^e 



Y 



y 



y 





DBVEI<OPMENT OF THE EQUATION 



.yCxy^' 



Q 



Fig. 



From a mathematical standpoint the sur- 

 face shown in Plate 6, A, presents the 

 fj problem of finding the equation of a surface, 

 x,yA given two sets of straight-line generators. 

 This can be done if the equations of any 

 three lines in the same set are known. 

 Select three Unes ah, cd, and ef (fig. 2). 

 Let %!, yi, Zi, and X2, y^y z^ be the coordinates 

 of two points upon line ab; x^, y^, z^, and x^, y^, z^ of two points upon line 

 cd; and x^, y^, z^, and Xq, y^, z^ of two points upon line ef. 

 The equations of the lines ab, cd, and ef, are 



(I) 

 (2) 



(3) 



and 



■^5 ^6-^5 2^5- 



From (2) the following equation for a plane perpendicular to the XY- 

 plane and containing the line cd is obtained : 



W4=(«- X3) O4 - ys) -(y-ys) (x^ - X3) = O. 



(4) 



Similarly from (2) the equation of a plane perpendicular to the YZ- 

 plane and containing the line cd is 



^5=(y-y3) (24- 23) -(^-23) iyi-y3)=o. 



(5) 



From (3), the equation of a plane perpendicular to the XV-plane and 

 containing the line ef is 



i(x-x,)(y^-y^)-(y-y,)(x,-x,) = 0. 



(6) 



Similarly, from (3) the equation of a plane perpendicular to the VZ-plane 

 and containing the line ef is 



Uj^(y-y^){z^-z,)- (z-z,)(y^-y^) = 0. 



(7) 



