1304 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



If w = I, we have: 





d'A 



d'A 



dudl 

 8/5 



-0, 



C' 



M3+^^3y. 



(37) 



^^(oC _B 



du" I \ I' 3. 



Substituting 



X 



'-4 



B ■ 



(38) 



we get instead of (33) 

 and instead of (37) 



x\dx'' - 1) - 2/ = 0, 



(39) 



d'A 



B 



"ar- = i«f(^^-i)- 



The positive root of (39) is plotted in Fig. 3. It follows that if r > 1 we 

 have a; > 1, consequently d^A/du^ > 0; and if r < 1 we have a; < 1, 

 consequently d^A/du < 0. Consequently if, and only if, r > 1 the 

 values u = ^ and x from Fig. 3 minimize the total bend loss A. 



Fig. 3 — Positive root of a;3(32;2 _ i) - 2r^ = 0. 



