STABILITY, PROPULSION, AND SEA-GOING QUALITIES OF SHIPS. 45 



developed in any individual instance, it is not necessary for the present pur- 

 pose to determine ; for it will appear that, whatever the excess may be, how- 

 ever abnormally, in virtue of it, the law of resistance for any given ship may 

 vary in terms of her velocity, a very simple scale of comparison wiU express 

 the relation between the excess as developed in a model and as developed in 

 a ship similar to the model when moving at a corresponding velocity. I 

 shall show, in fact, that if the velocities of the ship and the model are as the 

 square roots, these excesses of resistance thus arising wiU be as the cubes of 

 their respective dimensions, a law which, as is easUy seen, expresses also the 

 relation founded on those elements of resistance which vary as the square of 

 the velocity and as the squares of the respective dimensions. 



The principles on which Professor Eankine's stream-line investigations are 

 founded establish generally, in relation to all wholly submerged symmetrical 

 bodies moving in a fluid infinitely extended on all sides, that the stream-line 

 displacements which the motion of the body imposes on the surrounding 

 volumes of fluid are, for a given body, identical in configuration for aU velo- 

 cities (an identity which assigns to them always a velocity proportional to 

 that of the body itself), and that the configuration is similar for all similar 

 bodies. 



If we now suppose that the body is moving along the surface of the fluid, 

 and if we imagine the surface to be not under the influence of gravity or any 

 such force, it is obvious that here also the configurations of the stream-line 

 displacements wiU be identical at all velocities for the same body, and will be 

 similar for similar bodies, including those displacements which consist of 

 upward disturbances of the surface. 



When we impose the further condition appropriate to an existing water-sur- 

 face, that the replacements of the surface, when disturbed, are governed jointly 

 by gravity and by the volumes and velocities of the original impiilses of dis- 

 turbance, it follows that those impulses of disturbance, being similar for all 

 similar bodies at aU velocities, will retain their similarity wherever and in 

 the manner which the operation of gi'avity permits : and this will be when 

 the similar bodies are moved with velocities proportioned to the square roots 

 of their respective dimensions ; in these a similar wave-configuration will, 

 in each case, similarly dispose of the originally similar volume of displace- 

 ment, since similar-waves have their velocities so related. 



These waves (as is explained by Professor Eankine), when the velocity 

 proper to their length along the line of motion is exceeded by that of the 

 ship, so that they cannot squarely travel with her, satisfy the conditions of 

 their motion by travelling obliquely, and diverging into the surrounding 

 fluid, the angle of divergence and their size forming a measure of the work 

 constantly running away from the ship, and consequently of the resistance 

 caused by theii' generation. 



Xow the similarity of the configuration which has been asserted involves 

 the condition that when the velocities of similar ships are as the square roots 

 of their respective dimensions, the angles of divergence will be equal, and 

 therefore equal lengths of similar wave-crest will be " run ofi' " for equal 

 distances travelled by the respective ships ; and hence the energy abstracted 

 in each case by',thesc equal lengths of similar wave-crest is clearly as the cube 

 of the dimension (since the mass elevated is as the sectional area, and the 

 elevation is simply as the dimension) ; and since the forces which supply 

 proportionate amounts of energy while travelling a given distance must 

 be as the energy, it follows that the excesses of resistance thus called into 

 existence are also as the cube of the dimension, agreeing in this respect, as 



