1870.] 



Mathematical Theory of Stream-lines. 



209 



which ratio ranges in different cases from J to 1 ; of the acceleration and 

 retardation of ships as affected by the disturbance in the water, and espe- 

 cially of the use of experiments on the retardation of ships in finding their 

 resistance ; and of the disturbances of pressure which accompany the 

 disturbances of motion in the liquid. Up to this point the dynamical 

 principles arrived at in the fourth chapter are certain and exact, like the 

 geometrical and cinematic principles in the three preceding chapters. The 

 results obtained in the remainder of the fourth chapter are in some respects 

 approximate and conjectural, and are to a great extent designed to suggest 

 plans for future experiments, and rules for their reduction. These results 

 relate to the disturbances of level which accompany the disturbances of 

 motion when the liquid has a free upper surface, to the waves which 

 originate in those disturbances of level, and the action of those waves in 

 dispersing energy and so causing resistance to the motion of the vessel, 

 to friction, or skin-resistance, and the "wake" or following current which 

 that kind of resistance causes the disturbing solid body to drag behind it, 

 and to the action of propelling instruments in overcoming different kinds 

 of resistance. 



The resista nee en used bv viscositvis not treated of, because its laws have 

 been completely investigated by Mr. Stokes, and because for bodies of the 

 size of ships, and moving at their ordinary velocities, that kind of resistance 

 is inconsiderable compared with skin-resistance and wave-resistance. The 

 resistance caused by discontinuity of figure is stated to be analogous in its 

 effects to friction, but it is not investigated in detail, because ships ought 

 not to be built of discontinuous (commonly called "unfair") figures. 



Supplement. Received January 8, 1870. 



The author in the first place calls attention to the agreement between 

 the position of the points at which there is no disturbance of the pressure 

 on the surface of a sphere, as deduced from Dr. Hoppe's investigation, 

 published in 1856 (Quarterly Journal of Mathematics), and on the surface 

 of a short vertical cylinder with a flat bottom, as determined by the experi- 

 ments of the Rev. E. L. Berthon before 1850 (Proc. Roy. Soc. vol. v. 

 1850 ; also Transactions of the Society of Eugineers, 6th December, 1869). 

 The theoretical value of the angular distance of those points from the 

 foremost pole of the sphere is sin -1 1=41° 49' ; the value deduced from 

 experiment is 41° 30'. 



The author then adds some remarks on a suggestion made by Mr. 

 William Froude, that the wave-resistance of a ship is diminished when two 

 series of waves originating at different points of her surface partially 

 neutralize each other by interference ; and states that, with regard to this 

 and many other questions of the resistance of vessels, a great advancement 

 of knowledge is to be expected from the publication in detail of the results 

 of experiments on which Mr. Froude has long beeu engaged. 



