On the Equations which conduct to the Skew Surface. 54£ 



and introducing in the numerator of the last fraction Y— wZ and 

 X— mL for X and Y, we obtain 



A _JB w(Y-wZ)-n(X-wZ) 



X-a Y-/3 (X-«)(Y- ? ^Z)-(Y-/3)(X-mZ) , 

 which give the following values of X — AZ and Y— BZ, viz. 



Y _ A7 _(jE-roZ) [(X-«)(Y-nZ)-X(Y-nZ)-£X + «nZ ] - 

 (X-a)(Y-7iZ)-(Y-/3)(X--mZ) 

 (X-mZ)(/3X-aY) 

 ~ (X-«)(Y-»Z)-(Y-£)(X-mZ)' 



Y_-r 7 - (Y-nZ)Q8X-*Y) 



^-(X-a)(Y--nZ)-(Y--/S)(X-roZ)" 



Substituting these values in a former equation, we find 



(« 3 +/3 3 )(X-mZ)(Y-^Z)(/3X- a Y) 2 [(X-a)(Y-wZ) 

 - (Y-/3)(X-mZ)] -«/3[(X-mZ) 3 + (Y-rcZ) 3 ] (/3X-«Y) 3 =0, 

 or, by division with (/3X— «Y) 2 , 

 (a 3 + /3 3 )(X-mZ)(Y-mZ)[(X- a )(Y-^Z)-(Y~ / 8)(X-mZ)]-l __ 



-«/3[(X-mZ) 3 + (Y-rcZ) s ](/3X-aY) J ~°- 



Here the last factor can be altered thus, 



£X-aY = /9(X-«)-a(Y-£), 

 and thereby the equation put under the form 

 a(X-a)(X-mZ)[a 2 (Y-rcZ) 2 -/3 2 (X--mZ) 2 ] 



-/3 2 (X-a)(Y-7iZ) 2 [«(Y-wZ)-/3(X-mZ)] 

 + /S(Y- / Q)(Y-7iZ)[a 2 (Y-7iZ) 2 -/3 2 (X-mZ) 2 ] 



-a 2 (Y-/3)(X-mZ) 2 [a(Y-7iZ)-/3(X-mZ)] 

 This form puts in evidence the extraneous factor 

 «(Y-nZ)-/3(X-mZ), 



which, equated to zero, is the plane through the node and one 

 of the directing lines. Omitting this factor, we have the equation 

 of the surface in a new form, 



[«(Y-/iZ)-f/3(X-mZ)][«(X-a)(X-mZ) + /3(Y-/3)(Y~wZ)] 

 -/3 2 (X~«)(Y-7iZ) 2 -« 2 (Y-^)(X~mZ) 2 



Carrying out the multiplications, we find 



[a 2 (X-«) + /3 2 (Y-/3)] (X-wiZ)(Y~nZ) " 



+ a^(X-«)(X-mZ) 2 + «^(Y-/S)(Y-nZ) 2 l = 0; 



-/^(X-«)(Y-rcZ) 2 -« 2 (Y-/3)(X-mZ) 2 j 



}-»• 



