of Sound hy Spheroids and Disks. 373 



It will be convenient to write also 



M = cothf— Lsinhf 



N= 1 + 3 sinh 2 {■ - 3L coshf sinb 2 f 



(34) 



The final values of the coefficients become, after some 

 reduction, 



a = — -~ sinh 2 f cosh£< 1— — : . 12 cosh 2 f + 7 + 15Lcosh£sinh 2 £ > 



CO" 



3M 



4o> 4 



sinh f J 1— *gSj . 14L sinh f-4 coth f V 



jgg^ sinh 2 f cosh f (35) 



The next coefficient a 3 is of order co e ? and those succeeding 

 decrease continually by the factor or. a Q and a 1 are of the 

 same order, but a 2 is two orders higher. The orders have 

 been so retained that the result will be correct to order &> 4 . 

 Thus the summation in (32) is readily found to be, to 

 order g> 4 , 



7.2 3 



f = - — ^ (M sinh 2 f cosh f + sinh £ cos 0>- 1 '*' 



7cV sinh 2 £ cosh f f t , MtK . , 2 . . 1 nxr 4-12 cos 2 1 , 

 4 -7=77 - 1 Iocos-i9 + o4cosh 2 f— b — lON-i ^ \ e 



&V sinh £ cos J e 9 . . . 10 coth f ] ., , 0/ . s 



corresponding to an incident train <j>s=e tkz . 



The functions M and N depend only on the eccentricity of 

 the boundary, which is given by e = sechf. 



Thus T 1_ l + e 



Il= » l0 «T=S' 



MVl-^=l-|(l-^ 2 )log^' 



N,B-<3-2^)-|(l-. 2 )logg. . (37) 



When the obstacle is spherical and of radius p, the corre- 

 sponding result may be deduced by making c zero, and f 

 infinite so that 



|«*=/< (38) 



