THEORY OF COMPOUND CURVES IN FIELD ENGINEERING 141 



U = x 2 + y 2 , V = x 



U' = (x— a) 2 -f (y —b ) 2 , V' = x — yk — a+ bk. 



The equations of our special pencils of circles are therefore 



X 2 _j_ y 2 _ 2 \x = o, . (18) 



(x — a) 2 -f (y — b) 2 — 2 X' (x —yk — a -f- bk) = o. (19) 



Designating the co-ordinates of the centers of any two circles by (a, ft) 

 and («', /3') and their radii respectively by p and p', the condition for the 

 orthogonality of the two circles is 



Associating the values a, /?, p for a special value of X with the corre- 

 sponding circle of the pencil (18), we have 



a = \, ft = o, p = X. 



In the same way we associate the values a', /3', p' with the pencil 

 (19) and have 



a'= a + \',P' = b — \'k, p' = X' J(i 4- k 2 ). 

 Substituting these values in equation (5), there is 



(X — a — X') 2 + (\'k — b) 2 = X 2 + V 2 (1 -f k 2 ), 

 or, simplifying and solving for X', 



2 Xa — a 2 — b 2 



2a — 2 X — 2bk 



(20) 



According to this condition, to each value of X corresponds one and 

 only one value of X'; i. e., taking any circle of the pencil (18), there is 

 one and only one circle in the pencil (19) orthogonal to that circle. If 

 we substitute in formula (21) for X' and X successively the values 



b — £', 



X' = a' — a, X = a, and X r = 7 — , X = a, 



we obtain the two expressions 



2aa — a 2 — b 2 



a —a = 7i , 



2a — 2a — 2bk 



and 



k (2aa -a 2 - b 2 ) 



P —b = j 77- , (2l) 



2a — 2a + 2bk ' 



