of the Third Order. 515 



generating line will be 



X=#+AZ, 



Y=2,+BZ. 



When this meets the line (X— wiZ = 0, Y— raZ=0), we have 



mZ=#-fAZ, 



nZ=y+BZ, 



and thence 



x(n— B) — ?/(m— A) = 0; 



or, what is the same thing, 



nx — my — B# + A z/ = 0. 



And when it meets the line (X— a=0, Y— /3=0), we have 



«=# + AZ, 



£=y + BZ; 

 and thence 



B(a?-«)-A(y-£)=0. 



We have thus the system of equations 



X=# + AZ, 

 Y=y + BZ, 



nx — my — Bx -j- A?/ = 0, 



B(*-a)-A(y-|S)=0; 



from which, eliminating (A, B, x, y) } we obtain the equation of 

 the surface. 



Writing in the last equation 



B=»(*-«), A=«(y-/8) 



Rvalues which give B#— Ay=— s(/3#— ay)\ we find 



X + asZ = (1 4- sZ)#, 



Y-f-/&Z=(H-sZ)y, 



(n + /3s) # — (m + as)z/ = ; 



whence also 



(» + /&)(X + «*Z)-(?n + «s)(Y + /&Z) = 0, 

 that is, 



nX-roY+(na-m£)*Z+*(£X--aY)=0; 



or eUminating 5 from this equation and the two equations 



x— X -f- Z(x— a)s = 0, 



y-T + Z(y-/8)«=0, 



2M2 



