THEORY OF COMPOUND CURVES IN FIELD ENGINEERING 1 37 



/^H _ a 3 + V a 4 y / 6x + ^b 2 _ 6 3 + X' b 4 \» 



V i+X i+X' / + \ i+X i+A.'/ 



a, 2 + Xa 2 2 + & t 2 + X& 2 2 — r, 2 — Xr 2 2 



1 + X 



<z 3 2 + a 4 » + y + X- & 4 » -r 3 » - X' r A 



i+X' 



(10) 



& 2 2 -6 3 2 +r 2 2 +r 3 2 , 



V— Z> 3 2 + r, 2 + r 3 2 , 



V — V + r* 2 + ** 4 2 > 



V — V+^i 2 + V- ( JI ) 



Substituting the value for X' in (10) in (9), then for every value of X 

 the two corresponding circles with the equations 



U 1 + \U 2 = o (12) 



aX + b N 



u >+7xh u * = c (I3) 



are orthogonal. But according to (1) and (2) these quations (12, 13) 

 represent two projective pencils and their product is a curve with the 

 equation 



(aU 1 —b U 2 ) U 4 — {cU t — d U 2 ) U 3 = o. (14) 



As this equation is satisfied when simultaneously U 1 = U 2 = o, or 

 U 3 = U 4 = o, and as both pencils (12) and (13) pass through the cir- 



