no. IB.] APPENDIX. 143 



nitely deep (and if the difference of density Jq be regarded as infinitesimal) 

 <p is, with the same notation as before, 



a a - o 



9>(*) = e-^2— 2— • (6) 



The function cp is a holomorphic function for all finite values of a, with 



no zero for positive values of a if s > 1 and with only one simple zero 



a = ffj if s < 1. Just as in the case of the former problem we then obtain 



no waves if s > 1 ; and 



2.4 . ( x\ 

 h = — -^ — -, — j sin ff, „ for positive values of x, 

 Dcp (ffj \ D) 



if 8< 1, and if there is supposed to be no wave-motion ahead of the ridge. 



The resistance R is then calculated in exactly the same way as before, 



and 



e — S 2 



This quantity is represented by the broken curve in Fig. 3 PI. VI in the 

 same way as (5) was represented in Fig. 2 PI. VI. In the present case it will 

 be seen that the resistance is a maximum at below the maximum wave- 

 velocity (for v/Vm = - 77 about). When the vessel's velocity is further 

 increased to v m , the resistance decreases to nul. 



In Figs. 2 and 3, PI. VI the theoretical curves are drawn for comparison 

 alongside experimental ones, the latter representing as nearly as possible the 

 resistance due to wave-making. As the waves are assumed in the theoretical 

 calculations, to be created by a ridge stretching across the channel, while 

 a boat-model was used in the experiments, there can be no question of com- 

 paring the absolute value of the resistance according to experiments as well 

 as calculation. The scale of resistance is therefore simply chosen so that 

 the highest points of the experimental and of the theoretical curves shall be 

 at the same level. The scale of velocities is the same for both curves. 



The heavy full-drawn curve in Fig. 2 PI. VI represents the wave-making 

 resistance in a shallow canal, according to Scott Russell. For this purpose 

 the resistance given by Curve E in Fig. 4 p. 38 is diminished by a quantity in- 

 creasing as the square of the velocity, and which is at velocities above 8 miles an 

 hour, approximately equal to the total resistance given by E; the rest is assumed 

 to be the wave-making resistance and is represented as above mentioned, in 



