896 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



settled upon a method due to Whittaker, and this was used to complete the 

 calculations. 



Whittaker's method is described briefly in Whittaker and Watson's 

 "Modern Analysis," 4th Edition, p. 424. The method is developed more 

 fully in papers by Ince^. We shall confine ourselves here to some summary 

 indications of the nature of the method. 



The method leads to representations of the solutions of Mathieu's equa- 

 tion by formulae which differ in structure depending upon the part of the 

 (^0, ^i) plane in which we are working. We shall give the formulae suitable 

 for use in the neighborhood of the point ^o = 1, ^i = 0; the formulae for use 

 in other parts of the plane are given by Ince. 



Given the values of ^o and 6i, we first determine a number a by means of 

 the implicit equation 



el = 1 -\- di cos 2(7 + ^' (-2 + cos 4(7) - f^ cos 2(7 

 8 64 



+ 5Y2( ^- 11 cos 4(7) + 



12\3 

 Then we seek a solution of equation (3') in the form 



00 



\f/ = e'" Y^ {a2n+i cos[(2n + l)z - (7] + ign+i sin [(2n + l)z - (7]), 



n=0 



where ju, the o's, and the 6's are constants. We substitute the expression for 

 ^ into the differential equation, and determine values of the constants by 

 imposing the condition that the resulting relation shall be an identity in z. 

 After some rather intricate algebraic manipulations we finally arrive at the 

 following results: 



a, = 0, 61 = 1 



fl3 = T^ sin 2o- -f z-^ sin ^<^ + 04 ( ~ -9- sin 2(7 + 9 sin 6(7 j + • • • 



14^J . _ , uet . ^ , 



35^; . _ , 



^=(T08)(8^^^^2"+'-- 



^ Monthly Notices, Royal Astronomical Society of London: v. 75, pp. 436-448; v. 76, 

 pp. 431^142. 



