52 Proceedings of the Royal Irish Academy. 



substituting for the integral the product of (38) by + Jr + m-, there is 

 obtained for the first term the approximate value 



X [k cos(mr-mh+kz-k Wt)+{m-2Ctkr) sm(mr-mh + kz~k Wt)]/{k'+(m-2Ctkry | , 



(57) 

 or, replacing Ki(kr) by its approximate value 7r'^(2kr) ^ e^'', 



- I^(ka)K,(ki)(Uy'(k' + m') 



^[kco^{mr-mh^-kz-kWt)+{rii-2Ctkr)^m{mr-mh + kz-kWt^^^ 



(58) 

 Taking next the second term in the right-hand member of (29), it 

 may be proved by similar reasoning that under the same conditions, its 

 approximate value differs from (58) only in having the sign of the coefficient 

 of sin {mr - mh + kz - k Wt) changed. Thus, by addition, noting that in 

 (29) the second term in the coefficient of u is negligible compared with the 

 first, and neglecting k^ compared with ??^^ there results 



w = - m- cos {mr - ml + kz - k Wt)l2 { ¥ + {ni - 2Gtkr)-]. (59) 

 Caution is necessary in ascertaining from the equation of continuity the 

 corresponding value of u\ Owing to the rapid rate at which the second term 

 in the right-hand member of (29) varies with r, this term may have to be 

 taken account of. It may be shown that the approximate value of this term 

 is obtainable from (59) by changing the sign of k, and then changing the sign 

 of the whole expression. Eetaining the most important parts of each of the 

 portions of il\ we obtain* 



w = rii\m - 2Gtkr) cos {mr - mh + kz - k Wt)/2k [k' + (?» - 2Ctkry} 

 + m\m + 2Ctkr) cos {mr - mh - kz + k Wt)/2k {k- + {m + 2Ctkry] ; 

 or, substituting in the coefficient of the second cosine for 2Ctlir its approximate 

 value m, 



w =. m\m - 2Ctkr) cos (iiir - mh + kz - k Wt)/2k{Ir + {m - 2Ctkry j 

 + m. {4:1c)-' cos {mr-mb-kz + kWty (60) 



When m - 2Ctkr is of order k, the former of the two terms of (60) is the more 

 important; but when m-2Ctkr is zero, the latter; it does not follow, though 

 it may be shown to be true, that the second term is more important than the 

 omissions from the first; it does follow, however, that when m-2Ctkr is 

 of order k, vj and u are of the same order of magnitude, but that when 

 7/1 - 2Ctkr is zero, uiju is small. 



On comparing (59), (60) with (27), it is seen that, when m - 2Ctkr is of 



* Tliis deduction of the value of tv is not stiictly justifiable. "We ought to use the equation of 

 continuity to obtain an accurate expression for w from that given for « by (29), and then approximate 

 to its value. 



