LAMINATKD TIIANSMISSIOX LINES. II 1193 



totic foi'ins as r ^ » may be (locluccd by coiisidiM-iui;- the bchaA'ior of 

 //i(>) and Ii-At) i\)V large arguments, and arc 



Ai = A* :^ 1.169(2C)'^ + i\C - 2.025(20^1. (oCO) 



The magnitudes of both the real and iniaginaiy pai'ls of Ai and Ao thus 

 increase indefinitely with C. 



('i'^ f(^) ^ sinv!ioi(Jal function. Let 



f(t) = -cos2j.7r^, ^ ^ ^ 1, (5G1) 



where j/ = ^, 1, 2, 3, 4, • • • , so that/(^) goes through v complete cycles 

 in ^ ^ ^ 1. Then equation (532) reads 



(fw/d^~ + [A + iC cos 2j'7r^]w(^) = 0. (562) 



If we make the transformations 



W(t) = ui^), 



X = A/v IT , 



^ = -iC/2vV-, 



\\e get 



cf]V/dT- + [X - 2?? cos 2t]W(t) = 0, (564) 



and the boundary conditions (533) become 



Tr(0) = W(u7r) - 0. (565) 



Equation (564) is one of the standard forms of Alathieu's equation. 

 We are interested in solutions which are periodic with period 2 in ^, 

 and w^hich approach the form sin niT^ when C ^ 0. In terms of t and ?>, 

 the function corresponding to the ?nth mode in the Clogston line must 

 reduce to the form 



W(t) — sin - T. (566) 



.5—0 V 



For any value of ?^, this function may be denoted l)y' 



W(t) = sCmAr, t?). (567) 



(563) 



2' See N. W. McLachlan, Theory and Application of Mathieu Functions, O.xford, 

 1947, pp. 10-25, especially p. 13 and p. 10. In this roforence a or 6 corresponds 

 to our X, q to our t?, and v to our m/i>. 



