1 06 Mr Sharpe, On the Reflection of Sound at a Paraboloid. 



7. Under the above convention then the two solutions of (9) 

 in converging series will be found to be 



U^l-uA+^i^-l)--^^-^) 



?/ 4 

 + (1 ^ 2 (^-14^ + 9)-&c (15), 



U 2 = Uj logw + 2uA + ^ (- 3 A 2 + 1) + ^(ll^ s - 37 A) + &c. 



(16). 



The laws of formation of the coefficients of powers of u are 

 respectively 



(n + l) 2 a n+1 + Aa n + a n - x = 0, 

 (n + l) 2 b n+1 + Ab n + 6 n _! + 2 (w + 1) a n+1 = 0. 



[(13) when reduced to the symbolical form is what Boole calls 

 a trinomial equation, but if we put in it U = e iu U a + e~ iu Up where 

 U a is the same function of u and i which Up is of u and — i, it 

 will be found that the differential equation for finding U a is 

 binomial and the coefficients of the ascending series for U a and Up 

 are easily obtained. When this method is applied to equation 

 (12) I am inclined to think some interesting results might be 

 obtained.] 



The complete solution of (13) is of course 



U=BU 1 +CU 2 (17) 



where B and G are arbitrary constants. We should notice that 

 U-l = 1 when u = 0, and that udU 2 /du = 1, when u = 0. 



We shall call U x and U 2 the 1st and 2nd solutions respectively 

 of (13). The complete solution of (13) in Definite Integrals is 



U = - U*" A + 6 -^A I ^ cos (u co$d + A log cot g 



x [B + G log(u sin 2 6)] dd (18). 



This last equation is exactly equivalent to equation (17) the 

 exponential factor being necessary to satisfy the conditions that 

 Ui = 1 and ud U 2 jdu = 1 when u = 0. 



(18) is got by Laplace's Method : cf. Boole's Differential 

 Equations, Chapter xvin. For the exponential factor cf. De 

 Morgan's Differential Calculus, p. 669. Of course V in (12) can 

 be obtained from (15), (16) and (18) by changing u into v, and 

 putting — A for A. Similarly to the above we shall call F, and 

 V 2 the 1st and 2nd solutions of (12). 



