14 2 PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



the principal part of which is equal to the principal part of 



f pr-iS f7T+i I 



l ie~ <B+1/ys+ ~ sinh * \ e~*~ 8l " h 8 sin ' ' /2 *' d@\ + 6~ 2 " sillh * Bi " 2 1/2 * J c^^ i- 



U-.S Jit 



that is to the principal part of 



-2.- sinh t, sin* /,, 



2 

 o 



-21 1 



'}' 



and this is equal to the principal part of 



-r sillh 1 



_ /., . . f>-2 : sin l' * s1 " 2 '/a*i sift 



which is 



and therefore 



Hence, writing T = ~. sinh 8 (?i + ^)8 where z cosh 8 = 11 + -%, we have 



u = | (sinh 8)-''= e r , r = (sinh S)~"' ~ T , 



(x.). 



It remains to prove that as 8 hecomes small these expressions become identical 

 with those obtained for the case when n + ^z is of the same order as z' /3 . When 



8 is small 



z sinh 8 (n + J) 8 z sinh 882 cosh 8, 



that is, 



. 



' -v 



Now 



V- 2 {() 



and 



which is S~' 12 when 8 is small, and therefore as 8 diminishes, the form of the expression 

 in (x.) becomes that in (viii.). The approximations in (x.) can, as before, be carried 



