114 Lord Rayleigh. On the [Deo. 13, 



If the cylinder be complete, s is integral ; A s and B, are independent 

 constants, either of which may vanish. In the latter case the dis- 

 placement is in two dimensions only.* It is unnecessary to stop to 

 consider the demonstrations of (21), inasmuch as these equations 

 will present themselves independently in the course of the investiga- 

 tions which follows. 



It will be convenient to replace Sz, 8r, a0 by single letters, which, 

 however, it is difficult to choose so as not to violate some of the usual 

 conventions. In conformity with Mr. Love's general notation, I will 

 write 



The problem before us is the expression of the changes of principal 

 curvature and shifts of principal planes at any point P (z, 0) of the 

 cylinder in terms of the displacements u, v, w. As in (6), take as 

 fixed co-ordinate axes the principal tangents and normal to the 

 undisturbed cylinder at the point P, the axis of x being parallel to 

 that of the cylinder, that of y tangential to the circular section, and 

 that of normal, measured inwards. If, as it will be convenient to 

 do, we measure z and from the point P, we may express the undis- 

 turbed coordinates of a material point Q in the neighbourhood of P, 



(27). 



During the displacement the coordinates of Q will receive the 

 increments 



U) w sin + v cos 0, w cos 0-f v sin ; 

 so that after displacement 



y = 



or if M, v, w be expanded in powers of the small quantities z, 0, 



(28) ' 



y = a0 + i0o0 + v +T^*+-|-0 + .............. (29). 



az a0 



* See ' Theory of Sound,' 233. 



