500 



MH. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



2. The point P is defined before the strain by its a, ft, and lies on a certain 

 surface y = (the middle-surface). The prism whose centre is P is held in equi- 

 librium by the action of adjacent prisms, and its parts are not in the same configura- 

 tion as that in which they would be found if this prism were separated from the rest 

 of the shell and left to itself.* Now, if this portion were isolated from the action of 

 neighbouring portions, any point of it (Q) would take a certain position defined by 

 the intersection of three surfaces of the family (a, ft, y), which we may take to be 

 a -|- p, ft + q, r. Hence, when this prism is subject to the action of neighbouring 

 prisms the position of Q will be given with reference to the (x, y, z) axes at 

 P by pjh^ + u , q/h. z + % r/h a -f w , and after the strain is effected it will be given 

 by P/^i + u '> 9/^2 + v '> r /h s + w' referred to the axes of (x, y, z) defined in Art. 1. 

 The component displacements (u lt v lt w^) of Q are u u , v' v , w W . 



Consider a system of rectangular axes fixed in space, and after strain let , 77 , be 

 the coordinates of P referred to this system, and let the directions of the (x, y, z) axes 

 be connected with those of the fixed (, 17, ) axes by the scheme 



X 



y 



?n 



ra 3 



Then, after strain the coordinates of Q are 



0. > 



These expressions are functions of a + p, ft + q, r ; and, hence, for each of them we 

 have 3/t)a = 9/9p and 3/3/8 = 3/3g. In forming these differential coefficients it is 

 important to observe that ', v', w have no differential coefficients with respect to a, ft. 

 Throughout the space within which u', v, w exist, viz., the range of values of p, q, r, 

 which correspond to points within the elementary prism treated, a, ft do not vary. 

 In his theory KIRCHHOFF first introduces the differential coefficients analogous to 



* This remark was made by ARON, in his memoir in BORCHARDT'S (CKELLK'S) ' Journal,' vol. 78, p. 138. 



