PEOF. H. M. MACDONALD ON THE DIFFE ACTION OF 

 that is 



therefore <+x ~ 



n m (3/i) 2 tan |TT 1 .,_,, /0 , 2 



tan y = / i \ -TI , that is y = - 3 /! (3u) , 



' rj / i-i gg(j T^TT 4c 



whence 



<+X = '3 7r ~^~ y 3~' /2 (3/x.) 2 ( v i-) - 



When zn^ is of the same order as 2 1/s , the series for v iu in ascending powers of 

 /A is not suitable for obtaining approximate values, and must therefore be replaced by 

 a series involving inverse powers of fj.. To effect this it is necessary to obtain the 

 principal part of the integrals in (iii.) ; as in this case /A is negative, the principal part 

 is contributed by the second integral. Writing 



w = -3 M e-'>'-C, 



the important part of the integral arises from values of in the neighbourhood of 

 the value of that makes iv stationary; this value is given by = ( /j,) 1/! e~ 1/ < 1 '", and 

 substituting 



it follows that 



p~ l k^ \ p-z^l^-f jy _ -2i(-^-v 3 f ,,-<- ri l/ ' 2 e~ l/ *" l tf-(i*(Jir . 



O *-^^ ~~~ t' ^-vt > 



when p is not small the principal part of the integral on the right-hand side is the 

 same as that of the integral whose lower limit is oo e~ 1/6iri , and therefore the principal 

 part only being retained, 



(" l/ Sfl 



g-'/a-" g-3/ie <-^f^ = 3~V2 7J . 1 /2/_, t \-V. e -2 l (-M) / -'/>'_ 



Jo 



Hence, the principal part only being retained, 

 therefore 



and substituting for /A its value, this becomes 



. . . (vii.). 



These are the forms that the expressions for R and <f>, when z w 5- is of higher 

 order than z' 3 , take when a is near to ^ir ; for, writing ^TT a=e, it follows that 



= 2 COS a WTT+n + - a 



