180 -., ..,4^R» GRAVES ON A RECTIFICATION OF THE 



is a value of cos "' 0, which is always less than t (it being recollected that 6 

 is a quantity between 1 and — 1) it follows that the particular value of 



s r^ R l^3^..(2w-l)« ^ r >.2n + i 



vs« 



r^ R l^3^..(2;^-l)« /- r y + '-(^ j-^g. 



1.2 a/R" + S*"* 1.2...2«.2k + lAi/R^+SV "' J 



§ 6, In the equation a;- ^+ V-i b _ ^^^ ^^ determine what real values w may 

 possess, so that in each case a corresponding value of y may like- 

 wise be real. — [Note I.] 



.-1 



By [20], 



By [28], .,„aJhir:r','- ■•''/■•' 



- Q. 1 tr -L. n/\a 



f~ ^ = 2r7r + COS ^ -t=% — </ —ll \/ a^ 



a/ -r* 



Hence 



= f-(^(A+ A/-lB)(2/,r + c6s~' ;7p- \/-ll a/x";! 



or 



or (see [11] ). 



f-[ A(2j,r + c6s-'^) + BI\/P1 .f ^ v/~l [b (2/» + cos-'-;^-) _ Al i^**]} 

 In this expression the factor 



f { V~l [b (2/. + cc«-' ^JJ - Al ./J*]} 



is always real, as is evident on developing by [7] . 



Hence, that some y may be real, the other factor, viz. . 



must also have some real value. '■ - ' ' 



Hence (see [6] ) some one, at least, of tlirS '4«^iAities 



,,, „„ sin{A(2.> + c6s-'^)+Bl^p} 

 must = 0. ' , : 



