4.4 Proceedings of the Royal Irish Academy. 



A + Bf-, and the second part of the left-hand member of (7) disappears. 

 There can then be admitted no values of n, except such as make n + kW = 

 for some A^alue of r included within the tube. For the equation 



dHc 1 du u 



dr^ r dr r^ 

 being that of the Bessel's function of first order with a purely imaginary 

 argument; [di(k')')], admits of no solution consistent with the conditions 

 [requisite when the section is a circle], that ^<, = when r vanishes, and 

 also when r has the finite value appropriate to the wall of the tube [or 

 consistent with the conditions, which must be satisfied in an annular tube, 

 that u = for two real finite values of r]. But any value assumed by - kW 

 is an admissible solution for n. At the place where n + kW = ^, (8) need 

 not be satisfied ; and under this exemption the required solution may be 

 obtained consistently with the boundary conditions.* It is included in the 

 above statement that no admissible A^alue of n can include an imaginary part. 

 Lord Eayleigh then proceeds to consider disturbances which are unsym- 

 metrical. Taking u, v, iv, Q to be proportional to g'"(««+^-+««)^ the equation 

 which replaces (7) is highly intractable ; he shows, however, that no complex 

 value of n is admissible.f This result is also established when W is any 

 function of r whatever, provided 



d'W 1 dW k'r'-s'- 

 dr- r dr k~r' + s^ 

 is of one sign throughout the region. 



Art. 16. An Arbitrary Symmetrical Distiirhance resolved into a Series of Lord. 

 Eayleigh' s Type, vjhen Law of Floiv is that of Viscous Liquid in 

 Complete Pipe. 



It is seen from the above that there is only one law of steady motion 

 which can be fairly said to lend itself to an analytical investigation, and this 

 only when the disturbance is symmetrical ; this law, however, is at the same 

 time that which is of the greatest interest physically, as being that which 

 governs the steady flow of viscous liquid through a circular pipe, viz. : — 

 W = A + Br'^. Taking, then, this case, I proceed to show that Lord Eayleigh's 

 analysis suffices for the discussion of the most general disturbance, which is 



* As in the corresponding case of plane strata (Chap. I., Art. 1), Lord Eayleigh ohviously implies 

 that in the regions separated by the surface for which n + k TF vanishes, different solutions of (8) are 

 to be taken and tilted together so as to nialie u continuous. This, of course, necessitates slipping at 

 the dividing surface. 



t At this point in Lord Eayleigh's investigation there is a slight eiTor which does not affect his 

 conclusion. He regards (Collected Papers, in., p. 680, 1. 3) a certain function of r as a fixed number. 



