48 Proceedings of the Royal Irish Academy. 



where W = Cp- + C. I wish to show that under certain conditions the 

 ■^'alue of u given by this equation may at some times and places become very 

 great compared with its initial value. 



Consider first the former of the two integrals in (29), and of it the 

 portion which involves 7j^ + m^ as a factor, i.e. omitting this factor, 



- p sin m (p - I) sin 7: (,? - Wt) [I,(J:p)K,(J:lj) - I,(Ich)K,(J:p) dp]. (30) 



This may be written, as the difference of two, thus : — 



i {' p cos {m (p - hy-^k {z - Wf-)] [UJcp)X,(m - I,(Jch)K,{Jcp)] dp (31) 



-i[ p cos { m {p - b) - k (z - Wt) ] [Iikp)K,{M) - I,{kl)K,{kp)-] dp. 



(32> 



Xow the expression p [I,(kp)K,{kh) - L{kh)K,{kpy] 



is positive and increases continuously from zero as p increases from h ; for 

 Ii(oj) . increases continuously with cc, as may be seen from its expansion ia 

 powers of x, and xKi{cc) decreases continuously with increasing o:, as may 



be seen from the equation 



xK^(:x) = i [ e-^i''=^'i')ds. (33) 



Jo 



MoreoA'er, when x is large, /i(a-), -^i('-y) have the approximate values 



i,(a^) # (27ra3)-* e-, K,(x) = 7ri(2x)-^^-. (34) 



It seems evident, then, that if kr is sufficiently large, the most important 

 portion of the integrals (31), (32) is contributed by a small part of the range 

 near the upper limit, and within which the approximate values given by 

 (34) may be used. Considering first (31), making these approximations, 

 and noting that if k (p - h) is large, Ii{k'b)Ki{kp) is small compared with 

 Ii{kp)Ki(Jcb'), we replace (31) by 



{87TJi)'^K,(kh) I ' ph^p cos \mp - 7yib + kz - kt {Cp- + C')] dp. (35) 



i i 



In this, again, we may, without serious error, replace p' by r'^, so that under 

 certain conditions it is approximately equal to 



(SttA-) "* r^K,{klj) [' c^-'^ cos {//ip - rah + kz - kt {Cp'' + C')] dp, (36) 

 and this again to 



{87rky'r^K,{kh) 



\ kcos\rnp-ml:Hkz-kt(Cp'-\-C')}+(m-2Ctkp) sin [ mp-rnh^kz-kt(Cp-+C') \ j,.^ *• _ 

 -""I k'+{m-2Ctkpy ^'js' 



(37) 



