154 Journal of Agricultural Research voi. xii, No. 4 



at this point than would be the case if the indicated operations were first 

 performed with the symbols/ The general form of the equation resulting 

 from the previous operations is 



ax' + by^ -{-cz^ + 2fyz + 2gxz + 2hxy + 2lx + 2my + 2nz+d=^ O. (20) 



To reduce equation (20) to its simplest form the axes must be trans- 

 lated and rotated. 



TRANSLATION OF AXKS" 



The origin of equation (20) is translated to the center by putting 



x=x'+x„, y-y'+y'o, z+z'+z^; (21) 



he values of x„, yo, and Zg being obtained from the following: 



ax„ + hyo+gzo+l = (22) 



hxo + byo+lzo + ^n = (23) 



gXo+fyo+cz„ + n = 0. (24) 



These substitutions give, after dropping the accents from x', y' , and z' , 

 an equation of the following form : 



ax^ + fer^ + cz' + 2fyz + 2gxz + 2hxy + (7 = 0; (25) 



where G = /.ro + wj,, + w2o + rf. (25a) 



ROTATION OF axes' 



Equation (25) can be further reduced by a rotation of the axes. This 

 is accomplished by means of a cubic equation 



k:^-{a-\-b-\-c)k-' + {ab+ac + bc-p-g--h-)k-D = 0; (26) 



where D = 



a h g 

 h b / 



n j c 



(26a) 



Let the roots of (26) be A',, k.^, and k^. The desired equation, after trans- 

 lating and rotating the axes is 



k,x'+k,y!' + k^-^^^^^j^=0:' (27) 



' A numerical problem is ilcvclopeJ by this motho<l upon pages i,i6 lo i6o. 



' Snydkr, Virgil, and Sis/vm, C. H. analytic gbombtry oP spacu,. p. 77. Now York, 1914. 



'Idem, p. 79. 



Mdcm, p. 86. 



