50 Proceedings of the Royal Irish Academy, 



J. — Cylinder maintained full with orifice in middle of bottom. 



Let U denote the vertical velocity at the top of cylinder, w that at 

 orifice, both being supposed uniform over their respective sections. 

 Consider, now, two cylinders — one standing on the orifice and reaching 

 to the top of the vessel ; the other extending from this to the external 

 boundary of the vessel. Then, if the breadth of the cylindrical vessel 

 be considerable compared with its height, we may write for the cylinder 

 on orifice 



<f> = B,i- %B,,K,{nr) cos n% + Uz + ^^-^ (z^ - ir^-), 



where I is height of cylinder, n = s having all integer values, and 



3 is measured from the top. 



For the external cylinder, we may write 



<f>' = AQ-\- k log r + To (nr) cos 7iz + - ^ [z- - ^r^) . 



i^'ow, the Fourier expression for z is 



I 2^ 1 



^ = 2-7^^^^^^"' 



s having odd value only. The Fourier for is 

 , ^ 4 , 



Z^ = — + 2 ^ cos 711 COS 71Z, 



3 n^ 



8 having all values. 



• Also, may be regarded as the constant term in a Fourier's 

 expansion. 



We have, then, s differing from 0, and odd ; 

 . ^. p. 2^7 1 4Z7 1 



^ ^, 2^7 1 4{u- IT) cosnl 



= B,^K,in-R)- - -^-^ -2r' 



2 27 1 



for s even the term — will vanish from both sides. In general, 



C 7V 



therefore, we shall have 



A^Y,{nK) = B,,E,{nR) + cos nl ; (65) 

 also, A,,Y,{nE) = B,,K,{nR), (66) 



giving Bu = - ^di — r-^ cos 7ilYi{nIi), (67) 



^in' 



A,^ = -nB^ cos nlE,{nR). (68) 



