50 Proceedings of the Royal Irish Academy. 



which, again, is less than 



(43) 



If kr is large, this is known to be approximately equal to 



ekri2Jc'ri, (44) 



the neglect of which, in comparison with rM'yA', involves an error of 

 order l/h\ 



As regards the substitution of (35) for (31), the neglect of the term in 

 (31) which involves /, {kh) K^ {kp), is obviously valid. We have further 

 replaced /j (kp) by (27r kp)-^e^p, and the fractional error in so doing is known 

 to be less than Njkp where i\^is a definite number. Thus, this approximation 



involves an error of order not greater than the neglect of e^Ppljkp in 



comparison with r'^e^^'lk. The integral is less than k~'^ 



e^Pp''^dp, which is 



known to be approximately equal to e^''l'2k-r'^, and the error is thus of order 

 not greater than 1/kr. 



To sum up, then, the substitution of (38) for (31), which it is intended to 

 use, is legitimate when (i) m/k is large, (ii) kr is large compared with m/k, 

 (iii) the time is such that rji ^ 2kCt is small and of order not larger than k, 

 (iv) cos (nir-mh + kz-Wtk) is not a very small fraction, (v) k(r~b) is large; 

 and it will further be supposed below that (vi) k {a - r) is large. 



I next proceed to show that under the same conditions the term (32) is 

 negligible compared with (31), As before, by substituting the approximate 

 values of the /, K functions, and neglecting in integration /, {kh) K^ (kp) 

 compared with 7i [kp) Ki (kh), we obtain the approximate value 



(87^^■)" ^ ^i i^'^^) I p'e^' cos {mp - mh - kz + kt [Cp" +C')] dp, (45) 

 and this we replace by 



(87r/j)'^ri K, (kh) \ e^p cos { mp - mh - kz + kt (Cp" + C) j dp. (46) 



The approximations up to this point may be justified by reasoning similar to 

 that which precedes. 



Integrating by parts, the integral in (46) may be written 



\e^PQm{mp-mh-kz+M{Cp'' + C')] '' ['',... . , i^^n" n's^ 



' '^ V A^ /) <:^'9^Yi\[mp -mh - kz-vkt{Cp-+C)} 



m + 2ktCp 



^ b 



k 2kW 1 



m + 2Ctkp ^^^2Ctkpy\ ^^' ^ ^ 



