Manchester Memoirs, Vol. xlvii. (1903), No. 13. 3 



rigid body, it appeared that the effect of these forces 

 became almost neghgible at high speeds, and it was after- 

 wards determined to neglect the vertical motion of the 

 C.G. and assume the pitching small, since, according to 

 the experience cited above, no appreciable departure from 

 the actual case would be made. 



Proceeding in accordance with this assumption, the 

 shearing force at the ends of the vessel may either be 

 regarded as zero, when the solution may include a small 

 pitching motion,* or may have such a magnitude, with 

 due regard to sign, as to ensure that the boat shall be 

 horizontal and free from pitching. 



In the solution we have applied the first of these 

 conditions, but the application of the second, which 

 involves the neglect of a term which is small in com- 

 parison with other terms in the expressions, produces the 

 same stress at the middle section. 



In this analysis the manner of distribution of the 

 load has been assumed uniform throughout the length, 

 in order to simplify the work as much as possible. 

 Hence the problem finally resolves itself into one of 

 determining the stresses, in a uniform beam of hollow 

 rectangular section, uniformly loaded, and supported 

 throughout its length by a continuous distribution of 

 periodic forces, whose period is that of the waves 

 relatively to the vessel. Since every point of the beam 

 is subjected to a periodic force, the forced vibration will 

 not be of any simple character. From the result, how- 

 ever, it appears that the lowest critical period occurs 

 when the forced vibration is equal to the period of free 

 vibration of the bar with the ends free. 



The actual form of the wave surface will not be of 

 great importance, so long as the height and length are 

 * See Note, Appendix I. 



