96 DEVELOPMENT OF A CERTAIN INFINITE DETERMINANT ARISING IN [2 

 and 



90g" + 345" - 85(/ 4 - 1820<7 :i - 2668? 2 - 1440g - 288 



-v- T"\Y*OC!C!1 f\T\ ~~ -^ - - - - - -- . 



Also, 



in coefficient of ^ 768(/(r/ - ^ 1) (./- 4) [ " ^ + l} (<f ~ 4) 



In this way we see that no higher power of q 1 than the first occurs 

 in the denominator of the multiplier of sin qir. In the case from which we 

 have derived our equations, this is a small quantity. Hence the degree of 

 approximation attained by the above formula is most satisfactory. 



[It will be observed that in writing originally (p. 86) 



1 m! 



- 8 + ^ = l + ao+o 1 cosArt, 



we are ignoring orders of m above the second, corresponding to the fourth 

 order in our determinant. We now proceed to include the neglected terms.] 



Let the equation for z be 



d~z 

 = -r^- + z (1 + a + a, cos kt + a., cos 2kt + a. 3 cos 3kt + ...). 



Assume as before 



z = c a sin gt + ^ sin (g + k)t + c, sin (g + 2k)t+ ... 

 + c_ l sm(g k)t + c_,,sm (y 2k) t+ ... 



Then by equating to zero the coefficient of each sine in the result of 

 substitution in the above differential equation we obtain 



-a 1 c_ 1 --a^--a 3 c 1 --a i c, 

 c_ l --a 1 c --a,c l -- 

 0=... -2,c_ 2 --a 1 c_ 1 + [^-(l+a )]c --a 1 c 1 --a 2 c 2 - 



