94 



BIRKHOFF AND LANGER. 



are readily verified. Defining the determinants W x and W 2 by the 

 formulas 



(65) 



* ** 



Wx = | u& [$? ~ B eVi \ + «« {8if + S eVi } | 



* ** 



we have Z)(X) = D(X) where, in view of (64), D(\) satisfies the equa- 

 tion (56) and is given by 



(66) D{\) = [W M ] + [Wj\ e XT » w+B » w +[W 2 ] e XI V.W + «*»<» 



+ [Tf" ( " T) ] e x{r n (6)+r, 2 (6)}+ J B„ I (6)+B n (6) 



Sub-case B. arg I\(&) = arg { - I\ f (&)}. 



In this case #{Xr„ 2 (&)} has throughout S the sign opposite to that of 

 i?{Xr Vi (6)}, and it is found that 



;(»"•) 



= s(") _ S W 



(67) 



+ * w , = C = *£" + 5 C 



** 

 ft? - 5„„ ■ 



** ** 



Oce — Or.r. ~ 0, 



'cc v cc 



Accordingly we have 



+ [PFi]e x l r vl (b)+v^(b) }+B„ l (b)+B y2 (b) 



If now we define D(\) by the relation 



D(\) e - xr ^ b) - B ^ b) = D(\), 



it is readily verified on the basis of formulas (67) that D (X) again 

 satisfies equation (56) while it is given in this case by 



(68) D(\) = [W M ] + [Wj\ e xr ^ b)+B ^ b) + [W 2 ]e- XT "^ b) - B ^ b) 



+ [n* ( '"' ) ]r x f r '' l(6)_r " 2(6) I +B "i (6)_B ''2 (6) . 



Let us suppose now that the notation has been so chosen that 



r v (b) 



|r„,(6)|^ |r„(6)|, and set 



TAb) 



r. Then upon dividing 



equations (66) and (68) through by [W M ], (recall that W M 4= by 



