416 SURFACE WAVES. [CHAP. IX 



233. Rankine s results were obtained by him by a synthetic 

 process for which we must refer to his paper*. 



Gerstner s procedure f, again, is different. He assumed, 

 erroneously, that when the problem is reduced to one of steady 

 motion the pressure must be uniform, not only along that par 

 ticular stream-line which coincides with the free surface, but also 

 along every other stream-line. Considered, however, as a deter 

 mination of the only type of steady motion, under gravity, which 

 possesses this property, his investigation is perfectly valid, and, 

 especially when regard is had to its date, very remarkable. 



The argument, somewhat condensed with the help of the more modern 

 invention of the stream-function, is as follows. 



Fixing our attention at first on any one stream-line, and choosing the origin 

 on it at a point of minimum altitude, let the axis of x be taken horizontal, 

 in the general direction of the flow, and let that of y be drawn vertically up 

 wards. If v be the velocity at any point, and V Q the velocity at the origin, we 

 have, resolving along the arc s, 



dv dv 



&quot;5 = -** .................................... &amp;gt; 



on account of the assumed uniformity of pressure. Hence 



as in Art. 25. Again, resolving along the normal, 



v 2 1 d dx 



where 8n is an element of the normal, and R is the radius of curvature. 



Now v= -d\ls/dn, where \^ is the stream-function, so that if we write & for 

 dpfpd\lr, which is, by hypothesis, constant along the stream-line, we have 



tf dx 



Putting ^ ) 



multiplying by dyjds, and making use of (i), we obtain 



d*-x do dx _ dy 

 ds* ds ds~ * ds 



d 2 x dv dx _ dy 



J-9. &quot; J_ j_ &quot;&quot; &quot;jT&quot; \^/&amp;gt; 



whence, on integration, 



dx 



* &quot;On the Exact Form of Waves near the Surface of Deep Water,&quot; Phil. Trans., 

 1863. 



f &quot;Theorie der Wellen,&quot; Abh. der k. b ohm. Ges. der Witt., 1802; Gilbert s 

 Annalen der Physik, t. xxxii. (1809). 



