42 Proceedings of the Royal Irish Academy. 



clear what ratio we are neglecting ; it thus may he difficult to say how far 

 one approximation to a solution is good without obtaining a closer one or 

 the accurate one. It is, I think, reasonable to require the terms neglected 

 to be small compared with I (JJi- Uz), where ITi, U^ are the greatest and least 



values of ?7, instead of with / ?7. The term -—.——, which is neglected, is 



do: dy^ 



of order / 1 at the time when ni - ltd Uldy is zero ; neglecting this 



dif I 



(p JJIdii^ 

 term is thus equivalent to neglecting ' "i which would usually 



I [Ui— Ui) 



be of order 1/Pb'. Again, as to the terms neglected in {dldt+ U'djxd)V^4', 



with the approximate value of \p, which has been taken, 



d\p I sin {/ (x - Ut) + my} 



Isc^ ~ P + (m - ItdU/dyy ^ ^ 



d^ _ (m - ltd 17/ dy) sin { I (x - Ut) + my } 



. rm 4. m^A 



(60) 



dy P + (7/1 - ltd U\dyy 



^ 2 ltd'' U/df (m - ltd U/dy) cos {/ (x - Ut) + my] 



{l^+(m-ltdU/dyy-y' 



Apparently we may fairly neglect d^Ujdy- in d-^pidy and in the 



succeeding differentiations, if the second term in d\P/dy is small compared 



with the first, i.e. if ltd'' U/dy"- is small compared with P + (m - ItdU/dyy. 



and if we wish this to be so up to the time when m - ItdUjdy is zero, 



nuP Uldy''' 

 we neglect ,, , j,^, / , which is usually of order mlPl. I think then, 



* PdUjdy J I 



that under these conditions, the -ip of (58) may be taken as an approximate 

 solution of the equation (57). It, however, violates the conditions of vanishing 

 at the bounding planes y = 0, y = h. As stated in the previous Article, 

 there is reason to think that this objection might be ignored if mb be large. 

 Now, although this value of \p eventually decreases indefinitely, yet if 

 mfl is large, before decreasing it increases very much, approximately in the 

 ratio nP/P, the maximum value at any place being obtained at a time when 

 m - ItdU/dy is zero. There is thus, I think, evidence of possible instability 

 in the most general case and whether d'U/dy'' be one signed or not; I do 

 not of course regard this discussion as containing a satisfactory 'yroof. The 

 case for instability is further weakened by the circumstance that the critical 

 time at which i// is greatest now depends on y. 



