1888.] Bending and Vibration of tldn Elastic Shells. 



115 



dw 



_ 



cfa; 



dw 



_ 



(30), 



M , v , . . . being the values of u, v at the point P. 



These equations give the coordinates of the various points of the 

 deformed sheet. We have now to suppose the sheet moved as a rigid 

 body so as to restore the position (as far as the first power of small 

 quantities is concerned) of points infinitely near P. A purely translatory 

 motion by which the displaced P is brought back to its original posi- 

 tion will be expressed by the simple omission in (28), (29), (30) of 

 the terms w , , W Q respectively, which are independent of z, 0. The 

 effect of an arbitrary rotation is represented by the additions to 

 x, y, respectively of yO s 8 2l (^ x0 s , xO% yO- ; where for the 

 present purpose O lt 2 , 3 are small quantities of the order of the 

 deformation, the square of which is to be neglected throughout. If 

 we make these additions to (28), &c., substituting for a?, T/, in the 

 terms containing their approximate values, we find so far as the 

 first powers of z, 



du 



du 



dv 



_ 



dw 

 dz Q 



dw 



dv 



_ 



Now, since the sheet is assumed to be inextensible, it must be pos- 

 sible so to determine 1? 2 , 3 that to this order x = z, y = #0, = 0. 



Hence 



-0 



" 



dv 



The conditions of inextensibility are thus (if we drop the suffices 

 as no longer required) 



VOL. XLV. i 



