1 68 Journal of Agricultural Research voi. xii. no. 4 



We can find Xi, /Xi, Wj, the direction cosines of the angles which N makes 

 with the X-, Y-, and Z-axis in either of the following ways : 



If (20) (the equation of the surface of the plow bottom) is known, we 

 have by differentiation 



Xi ^^1 



ax^ + byo + gzo + l hxg + byo + fzo + m 



= ^1 _ (rA) 



gxo+fyo+czo+^i 

 I 



v 



l(aXf, + by^ + gz^ + lf+ {hx^ + by^ + fz^ + mf' 



or if the paths of the soil particles are known but the equation of the 

 surface is unknown the angle A^^- can be measured by means of a pro- 

 tractor and plumb bob, as shown in Plate 9, C. The direction cosines 

 Xi and Ui can then be computed from the following : 



(Xi)'+(mi)^+ (1^1^=1 (55) 



^ dx ^ dy ^ dz ^ . ^, 



where the values for ^, J-, and -y. can be obtained from (49), (50), and 



(52). 



, dx dy , dz 

 The direction cosines of T (X^, yUj, 1^2) are proportional to ^, ^-, and ^' 



Hence 



dx dy dz //^Y , /^V + /^— V (57) 



dt dt dt '\\dt)'^\dt)^\dt) 



The direction cosines of T (X3, ^3, v^ can be computed from the follow- 

 ing:^ 



(X3)'+(M3)'+(i'3)'=I- (58) 



^' ^ ^^— =±i. (59) 



M3l^2 ~ l^3M2 /^3«^1 — l^sMl Ml 1^2 — l'lM2 



The components in the directions N, T, and R of the forces acting on a 

 soil element of mass M, moving with the component accelerations 

 d^x d^y J d^z 

 W df ^^^ dt^ ^'^ 



d^x d^v d^z 

 Fn = M(\^^+^,J + v,-i^,) (60) 



FT-MCX.g^+^f+v.lf) (6.) 



F. = M(X^ + ,,^, + v,^). (6a) 



' Snyder, Virgil, and Sisam, C. H. Op. cit., p. 40. 



