88 BIRKHOFF AND LAAGER. 



Z)( X ) = I Wa ([*«]) + W b ( C xr ; (6)+B ; (6) [5 l7 ]) |, 



i.e. 



(49) B{\)=\W^} + [u^\c^^^\. 



In this as in subsequent formulas when the expressions in question 

 are determinants the letters r and c rather than i and j are used to 

 indicate row and column respectively. 



Due to the fact that those and only those values of X which satisfy 

 equation (47) are characteristic values, i.e. values which yield solu- 

 tions of system (46), equation (47) is known as the characteristic 

 equation of the system in question. 



The introduction at this point of quantities 8^ and <5,-y will do much 

 toward simplifying the further discussion. These quantities are de- 

 fined by the relations 



( = 5u when RiWjib)} ^ ( = when iJ{xr,-(6)} ^ 



( = when RlW^b)} > 0, ! = 8 {j when #{xr y (6)} > 0. 



It is to be noted that 5# and 8jj are functions of arg X alone. 



Suppose now that the rays 9 R{\Ti(b)} = 0, i = 1, 2, . . .n, are 

 drawn in the plane of the parameter X and let one of the sectors 



bounded by a pair of adjacent rays of this kind, jR {xr *.-(&) } = 0, and 



* ** 



R{\T l (&)} = 0, be denoted by <x u . Then if 5 ( t *° and 5 ( ,*° are respec- 

 tively the dij and 5;* for some (any) particular value of X within <r w , 



* ** 



we have the identities 8 fj = 5 ( *j\ 5, 7 = 5^-', for all X in the interior of 

 the sector in question. It becomes apparent in virtue of the relations 



' [w%] when R{\T c (b)} <0 



(W) \w {a) ] 4- f?r (b) l Ar c (b)+B c {b) = J 



^ou; lm rc j -r L« rc J e | r w (fe)j e xr c (6)+B c (6) 



whenit{\r e (6)} > 0, 



and the fact that R\\Ti(b)} 4= 0, i = 1, 2,. . .n, that Z)(X) takes, for 

 any X in the interior of such a sector the form 



8 A half-line issuing from the origin of the X plane will be referred to as a ray. 



