Mr. A. J. Robertson on the Positive Wave of Translation. 521 

 and 



=± \/^- • • • <*■> 



a 



from which results the following value for L : 

 From (7.) and (8.) we also obtain 



the equation to the curve in terms of x. 



dy _dy dt __ irk . ,-./,.* . ,- \ / m n 



dx-Tt'Tx- LTV • sin ^ V + « versm *>•'./• (11 ° 



When £=0 or =T, -p = 0, or the tangent to the curve at 



the ends and at the crest of the wave is horizontal ; therefore 

 the curve at the ends is convex, and at the crest concave, to 



the line of repose. Now g ~ would be the accelerating force 



on the supposition of the whole of that force being expended 



in producing forward motion. But the cause of both vertical 



and horizontal motion is the slope of the surface, the amount 



d\i 

 of which is expressed by — ; and it follows that the definite 



ace 



integral of g . -~ between the limits a? = and x = L represents 



the sum of all the forces producing forward motion, and of all 

 the forces producing vertical motion. Hence we obtain an 

 equation from which the value of the unknown quantity //, 

 may be deduced. 



The space 5 which a column has moved through in a hori- 

 zontal direction in time t, is 



l+ 1)t~ x 



L+I 



^{'-^x/ctHa/^-^ 



