PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



which n + %z is of lower order than z 1/3 ,* the terms for which zn% is of the 

 same or of higher order than z 1 * will now be considered. Denoting these terms 

 of the series S 3 and S 4 by S 32 and S 42 , and remembering that 



where T is a negative real quantity increasing rapidly with n, 832 and S 42 are to the 

 required order of approximation given by 



S 32 = - (27r)-' /5 (xr, 2 )- 1 sin 1 ' 9 t (n + ) % sec 1 '- a sec'/* a^ n -'< " "-<" +1/ *> (+.->-v^ ; 



r-A.-'/ 



S 42 = - (27r)-'- (/c)', 2 )- 1 sin'- 2 (n + ^ sec 1 /' a sec % ai e' [ "-* cos a ~ r ' cos a '- ( " + ' (-+-.+0+ 1 /,-]. 

 The value of-- {mrz cos a 2j cos a. l (/'+|-) (a + otj ^)} is irot^ + 0, which, 



<-Vlt/ 



since a and j are less than ^TT cannot be very small, and therefore the sum of the 

 terms S 33 is of lower order than ^ Again the value of 



-j- {UTTZ cos a 2! cos <*! (n + ^) (a + i + 0)} 



(.til t> 



is TT a a, #, and, if P is a point outside the geometrical shadow for which the 

 angle OPO, is less than a right angle, there is a value vt, of n in the series S 42 for 

 which Tr a. a l vanishes, viz., that given by i + |- = Kp where p is the perpendicular 

 from on OjP, and the principal part of S 43 is given by 



S 42 = -(27r)-"-' ( K r*Y l (v^ + i) -'- sec"-' a sec- a^-0 +* tV 



where 



vtj + ^ = g sin a. = z l sin a,, 

 that is by 



S 42 = -(27r)--(Kn 2 )-V^44)'^^ 



\z cos a Zj cos 

 or 



S 42 = IK sin 2 

 where D is the distance dP. Hence 



The terms of the series S ls S 2 , S 3 , S 4 for which I'/i + g z\ is of lower order than z k and 

 z\n\ is of the same or higher order than z^ k are of the order z''^, and, that their 



* It will be proved below that these terms are important only in the neighbourhood of the geometrical 

 shadow. 



