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 the velocity at the origin, we 

 have, resolving along the arc s, 



on account of the assumed uniformity of pressure. Hence 



a = V- 2 5^ (") 



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



v* ldp_ dx 



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



Now v -d-^/dn, where ^ is the stream-function, so that if we write cr for 

 dp/pd\lf, which is, by hypothesis, constant along the stream-line, we have 



v 2 dx 



Putting IIR = --^ 



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



d 2 x dv dx_ dy 



whence, on integration, 



dx 



* "On the Exact Form of Waves near the Surface of Deep Water," Phil. Trans., 

 1863. 



t "Theorie der Wellen," Abh. der k. bohm. Ges. der Wiss., 1802; Gilbert's 

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



