I y4 Journal of Agricultural Research voi. xii. No. 4 



moved backward, the straightedge remaining parallel to the plane 

 z=0. By taking the point o as the origin, the equation of the surface is 



T,hyz—2dxz—2hly + 2hdz=0'^ (82) 



h = breadth of furrow 

 c^= depth of furrow 

 / = length of moldboard. 



On rotating the XV-axes through tan-^= 2^/36, the equation is 



{()h^' + J[(P)y'z-^hdlx'-6bHy' + 2hd^Jghi'+^(Pz=0. (83) 



On rotating the yZ-axes through tan-^>^V2, the equation 13(96^ +4^^) 

 {iy"f-{z'Y\-?>hdlx' 



^-2{hd^|^WT^-2>hH^fi)\y" + z']=-0, (84) 



Translating the axes to the points 



y =>' +yo 



z' = z" +Zo 



where yo has such a value that 



2{9f^ + 4(P)yo+ 2[bd^iSb^+Sd^- 3m-yf^]=0, (85) 



and Zo has such a value that 



-2(9b^ + 4d^)Zo+2[bd-yJi8b' + 8d^-3m^]=0, (86) 



gives 



(9b^+4d^)[(j"'y- (z'y]-Sbdlx'+ (>'o'-2o')(9&' + 4cP) 



+ (yo+Zo) {2bd^iSb^+Sd'- 2,m4i) =0. (87) 



Letting the constant terms in (87) equal C gives 



(962 + 4d2)[(y'")2- {z"y]-2>bdlx' + C=0. (88) 



Translating the axes to the point x' = x"-\-Xo where Xq has such a value 



that 



-8bdlXo-VC=0 

 gives 



(962+4<P)[C|/'")'- {z"f] = 8bdlx". (89) 



This is the equation of a hyperbolic paraboloid.^ 



IvAMBRUSCHINl'S PI.OW BOTTOM 



Lambruschini,^ an Italian, describes a method for generating the sur- 

 face of a plow bottom which he considered to be more efficient than 

 the surface developed by the Jefferson method. Lambruschini proposed 



» The method of developing the equation for this surface is given upon pages 150 to 156. 

 *Snyder, Virgil, and Sisam, C. H. Op. dt., p. 73. 



» lyAMBRUSCHINl, R. Op. cit., p. 37-80. 1832. 



