of Sound by Spheroids and Disks. 375 



Thus L=log^-g 2 , (41) 



and ultimately, the first order effect is 



_^ 2 a + cog ^_^ _ ^ ^ (42) 



the higher order being readily calculated if required. The 

 effect is zero at the extremity nearest to the incident waves, 

 and a maximum at the other. This was to be expected. 



Diffraction by an oblate spheroid. 



In this problem the. substitution p + iz= c cosh (a 4- ifi) may 

 be employed. The equations so obtained for the harmonics 

 and associated functions are identical with those for the case 

 of the prolate spheroid, if the sign of a> 2 be changed, and if 



/ LIT 7r\ 



(a, /3), in the latter, correspond to (a— -- , ft— , ) respec- 

 tively in the former. 



L thus becomes, for the oblate spheroid, 



i tan -1 cosech f -f isir, 



and it is readily seen that s = 0. 



The expression for the effect of an oblate spheroid on 

 waves incident along the direction of its axis is then readily 

 deduced from the previous case of the prolate obstacle. 

 Employing now the notation 



L = tan" 1 cosech f, 



M = cosh f tan -1 cosech f — tanh f, 



N = 1 — 3 cosh 2 f + 3 sinh f cosh 2 f . tan -1 cosech f, . (43) 



it appears that, corresponding to an incident wave, 



</> = ^, 

 there is a divergent system 

 k 2 c z 



i ~ li ( M sinh ^ cosh 2 ^ + cos e cosh & e ~ ikr 



+ ^sinhf cosh 2 ^10N-6cosh 2 f-15cos 2 ^-48- ~ . 1 - 3 cos 2 V«* 



&V cosh £ cos 6 /.. . „ 9/1 10tanh£\ .. > . JN 



+ 150,-M (U-aco S -g + ^ r -^)>-^. . (44) 



