1889.] 



Presents. 



49 



change of curvature is large in comparison with the extension, except 

 at poiuts in the neighbourhood of the edge, where a w — 0) is com- 

 parable with h. 



It is also shown that the tension T 1 parallel to the axis, and the 

 couple G 2 about a circular section do not vanish at the circular edges, 

 but have finite values ; and therefore a tension and a couple of the 

 proper amount, which tends to produce synclastic curvature of the 

 generating lines must be applied at the circular edges. If, therefore, 

 this force and couple were removed, anticlastic curvature of the 

 generating lines would be produced, and this would involve extension 

 of the middle surface parallel to the axis. It is, however, obvious 

 that a thin shell, under these circumstances, does not assume a saddle- 

 back form, and therefore the anticlastic curvature, and the extension 

 upon which it depends, must be exceedingly small, except in the 

 neighbourhood of the circular edges. 



The difficulty of satisfying the boundary conditions at a curved free 

 edge, when the middle surface is supposed to be inextensible, partly 

 arises from the fact that it is impossible for the flexural couple about 

 the curved edge to vanish, unless some extension or contraction takes 

 place in the neighbourhood of the edge ; but the inference to be 

 drawn from the statical problem considered above is, that when a thin 

 shell, whose edges are free, is vibrating, the amplitudes of those terms 

 upon which the extension depends are small in comparison with the 

 amplitudes of those terms upon which the bending depends. More- 

 over, a variety of results which have been obtained during recent 

 years indicate, that the pitch of notes which depend upon extension is 

 very high, compared with the pitch of notes which depend upon 

 flexure ; and this circumstance, combined with the smallness of the 

 amplitudes of the extensional vibrations, points to the conclusion that 

 the former notes are usually feeble in comparison with the latter. 



The values of the edge stresses and the equations of motion are 

 also obtained for a spherical shell, but the work is the same as in the 

 case of a cylindrical shell, except in matters of detail. 



The Society adjourned over the Christmas Recess to Thursday, 

 January 9th, 1890 : 



Presents, December 19, 1889. 



Transactions. 



Kalle :— Verein fur Erdkunde. Mitteilungen. 1889. 870. Halle 



1889. The Verein. 



Helsingfors : — Finska Vetenskaps-Societet. Acta. Tomus XVI. 

 4to. Helsing for sice 1888 ; Of^ersigt, 1887-88. 8vo. Helsing- 

 fors 1888. ' " The Society. 



VOL. XLVII. E 



