370 Dr J. "W Nicholson on the Scattering 



With the integrals o£ the squared harmonics the coefficients 

 become 



A = 1 + ^ (1 - 3 cosh 2 a) + ||(cosn 4 * + 2 cosh 2 « + j^j '+... 

 1 + ^- (3 — 5 cosh 2 a) + ™r(cosh 4 oc 



A 2 = ^ (1-3 cosh 2 a)+ 



A 3 =-^(cosh 3 a-f cosha)+ (23) 



Associated Functions of the Second Type. 



It may be proved in the usual manner that a second solution 

 of the equation (3) tor iv n is given by 



vJ*) = w I -v-r^ =•, .... (24) 



where /3 is any root of v n (j3) = 0. 



In calculating the functions v n near the obstacle, we may 

 treat /? as approximately infinite, the integral between any 

 two possible values of /3 being zero. 



Thus 



vM = r°°-^_ + -: ( i + 3cosh 2 a ) f" *?- 



v y sinh a 18 J smha 



a> 2 f 5 l + 3cosh s « 7 



+ -q ^u ^ a 



y smn a 



= log coth - + — I — 6 cosh a + 1 — 3 sinh 2 a log coth - j. 



An additive constant depending on ft only is ignored, for 

 the functions v n will all be subsequently differentiated. 

 {Similarly 



Vi(ol) = j ( cosh a . log coth o — 1 1 



+ -rA 6 + 5 cosh 2 a— 11 cosh a + 5 cosh 3 a log coth ~ I f (25) 



on reduction, it being unnecessary to ignore any constant in 

 this and subsequent cases. 



