116 Lord Rayleigh. On the [Dec. 13, 



du dv du dv 



= 0, w + = 0, r + aj- = .. . (31), 



dz d<j> d(j> dz 



which agree with (21). 



Returning to (28), &c., as modified by the addition of the trans- 

 latory and rotatory terms, we get 



x = z + terms of 2nd order in z, 0, 

 y = 



dv 



<%, 



or since by (31) d^w/dz^ = 0, and dv/d<j) = w, 



The equation of the deformed surface after transference is thus 

 1 dv 1 d z w 1 , f 1 1 1 d*w 1 



Comparing with (7) we see that 



1 1 1 / d*w\ l/dv 



so that by (19) 



This is the potential energy of bending reckoned per unit of area. 

 It can if desired be expressed by (31) entirely in terms of v.* 



We will now apply (24) to calculate the whole potential energy of 

 a complete cylinder, bounded by plane edges z = +Z, and of tbick- 



* From Mr. Love's general equations (12), (13), (18) a concordant result may be 

 obtained by introduction of the special conditions 



*! = 0, Aj-l/a, l/pi = 0, l/p 2 = I/a, 



limiting the problem to the case of the cylinder, and of those 



<7j = ff 2 = or = 0, 



which express the inextensibility of the middle surface. 



