310 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Moreover, if <^ (7") is any polynomial in T, we have 



In particular, if ^^ = ± 1 , 



F, {T-') ^ F, {T) ', 



and if 4 4^ ± 1> and 4' = 4"^ has the same multiplicity as 4? then 



F,> {T-') = F, {T) . 



"We have now the following theorems : 



I. If t^is real and T^ e^^^ is also real, then T^ = 1* 



II. If T is real and no negative number is a root of the characteristic 

 equation of T, there is a real polynomial /"( 7") satisfying the condition 



III. If T is real and each negative root other than — 1 of the char- 

 acteristic equation of T is paired with its inverse, there is a polynomial 

 f(T) satisfying the equation T= e-^'^' and such that/(r) —f{T-'^) is 

 real.f 



* For if T = e^^ is real, then 



61 o! 



is real. Therefore, if U is real, T-T-^ = 0; that is T'^ = 1. 



t If J" is real, each imaginary root of the characteristic equation of T is paired 

 witli its conjugate imaginary. Therefore, if this equation has no negative root, 

 f(T) may be taken real by a proper choice of mj, jh.,, etc. 



If the distinct negative roots of the characteristic equation are 



and C2,/4-i — ~ ^> ^""i moreover (for i = 1, 2, . . . v') d ^^^ Cf^ have the same 

 multiplicity, then, for a proper clioice of m^, m.2, etc., the imaginary part of the 

 polynomial /{J") is 



n /^l Si"' (Fi (T) + F,, + i (T)) + n V'^F^.'^i {T) ; 



and, since now 



Fi ( r -1) = F,,+i { T), F,,+i (T-h=Fi{ T), 



{i = 1,2 /) 



F2u'+i(T-') = F,,,^l{T), 

 therefore/(r) -/{T~^) will be real. 



