Jan. 28, i9i8 Study of Plow Bottoms 153 



Consider 



Ui = Au^. (8) 



where A is a constant. This is the equation of a plane which contains 

 the intersection of planes (4) and (5) ; hence it contains the line cd. 

 Similarly 



u^ = Bu^ (9) 



where 5 is a constant, is the equation of a plane which contains the line ej. 

 If A and B have such values that the point %', y' , z' is on (8), (9), and 

 (i), the line of intersection of (8) and (9) meets (i) and is a generator 

 (see fig. 2). Hence, 



_ {x'-x^) {y- y^) - {y'-y^){x- x^) . 



{y'-y,){z-^,)-{z'-^^)iy.-yz)' ^ ^ 



_ {x ' - Xs) (ye - ^5) - ( :^ ' - yo) K - x^ . 



iy'-y.)i?.-^.)-i?'-z,)iy,-y,)' ^"^ 



and 



x'-x^ y'-yi z'-Zi 



= K; (12) 



«2-«i y2-yi 22-2i 



where 7v is a constant. 

 From equations (12) 



x' = K(x2-Xi)+Xi (13) 



y' = K(y^-y,)+yi (14) 



z'=K(z2-Zi)+Zi (15) 



From equations (10), (13), (14), and (15) 



i[K{x.2-Xi)+x^-X3](y^-y3))-([K{yr.-yj)+y^- y3](x^-Xs)) ^ ,^. 



ilK(y2-yr)+yy-y3]{Zi-h))-i[K{^2-2i)+^i-h]iy*-y3))' ^ ' 



and from equation (8) 



J^ JKy■-^■^{y^-y^-{y-y■^{'^^-'>^ . 



(y- J'3)(24 -%) - (2-23)(j'4-3'3) 



From equations (11), (13), (14), and (15) 



(17) 



(\K{Xi-x^-\-x^-x^{y^-y^y)-{\K{y.2-y;)-\-y^-y^{:x^-x^) ^ , ^. 



^~ ^\K(y.,-y,)\y,-y^{z^-z-^)-{\K(z^-z,nz,-z^\y^-y^)r ^'""^ 



and from equation (9) 



jj_^ (^-^5)(y6-y5)-(y-?'5)(^6-^5) t \ 



{y-yC){2t-z^)-iz-zi){y^-y^) ^' 



Eliminating A^ B, and K from (16), (17), (18), and (19), we have the 

 equation of a surface through the lines ab, cd, and r/. The equations are 

 left in this form because numerical substitutions are more easily made 



