46 Proceedings of the Royal Irish Academy. 



For a disturbance varying as e**'', each element of the integral in (15) is 

 then to be multiplied by g-*'^''^*. And the result obtained is that, if initially 

 the disturbance is given hj u = i^'^ f{f), then, at time t, it is given by 

 - {/, {ha) K, (kh) - /, (kb) 7C (ka)] u = e'''' [I, {kr) K, (ka) - I, {ka) K, (kr) ] 



{(pk' + n-')Ap) -np)-pf\p)] U.{kp)K,{kh) -I,{kb)K,{kfj)]e-^''^'dp, 



- another term obtainable from this by interchanging a, h, (16) 



the argument in W being p. 



Art. 1 7. The preceding Besidt obtained more directly. 



The result to which this analysis would lead may, however, as in the plane 

 case,* be obtained more directly from the fundamental equations. Eliminating 

 Q from (1), (2), we obtain 



fd ^,^d\fdiv dtC\ fdw du\dW d^W ^ ,^^. 



this equation is, as far as terms of the first order of small quantities, the 

 equivalent of 



fd ^ f-[j;r , ,„\ ^ , d d(W+iv) ^ dn\[d{W + w) dii 



which expresses the constancy of the vortex strength.! By using (3), 

 (17) becomes 



Ul ^^d\(dw du\ fd'W ldW\ ^ 



^dt dzj \dr dz I \ dr^ r dr 



and, again using (3), we obtain 



d d\ id^u 1 du u d'^ti) dufd^W 1 dW\ _ , „. 



ydt dz J \ dr"^ r dr r^ dz^ ) dz \ dr"^ r dr ) 



an equation of which (7) is a particular instance. For the form of W with 

 which we are dealing, the second part of the left-hand member vanishes ; and 

 we obtain as an integral 



d^u 1 du u d^u _ ,„... 



dr"^ r dr r^ dz^ \-^ > J> 



where F may be any function, but is determined from the initial values of u. 

 If we now introduce the supposition that u SiS, a. function of z is pro- 

 portional to 6**-, (21) becomes 



p^ + lp.l jcHo^e-^^^^fir). (22) 



dr^ r dr r^ 



* Chap. I., Art. 5. 



t Vortex strength is not vorticity, but proportional to the product of vorticity and sectional area 

 of the vortex filament. 



