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SCIENCE. 



[N. S. Vol. VIII. No. 202. 



wave problems. Whether much more can 

 be done by the present analytical methods 

 is not certain. We may frequently meet in 

 this department with series which formally 

 satisfy the equations, but which either con- 

 verge very slowly or not at all. The 

 theory of free and forced waves, depending 

 to a large extent upon methods of approxi- 

 mation, when numerical results are re- 

 quired for comparison with observation, is 

 in many respects fairly satisfactory if the 

 height of the wave crest above the mean 

 level is not very great and if the motion is 

 slow. But there are comparatively few 

 problems solved for shorter waves in con- 

 fined bodies of water. The oscillations of 

 water across a canal of circular section is 

 an instance of a problem which should yield 

 to analysis. But little is known of the 

 causes which produce double and other 

 curious kinds of tides in enclosed areas. 



Another class of problem is the wave- 

 resistance experienced by various shaped 

 bodies moving over water, especially when 

 the velocity is not small. That this class is 

 becoming extremely important, in view of 

 the high speeds attained by torpedo boats, 

 there can be no doubt. The lines of a ship 

 are at present a matter mainly of experi- 

 ence. One of the most puzzling things is 

 the difiference in the speed qualities of 

 two ships built on the same plans. A very 

 slight difiference in the lines will frequently 

 enormously alter the speed qualities, and 

 one can scarcely doubt that this is due to 

 the differences in the waves which are 

 formed as the ship moves. The kind of prob- 

 lem thus presented to the mathematician 

 is always a difl&cult one. Briefly stated, it 

 is a problem where a slight difference in 

 the boundary conditions makes a large dif- 

 ference in the eflTects produced. A similar 

 difficulty occurs when one approaches the 

 limits of stability of steady motion. 



The treatment of viscosity and skin fric- 

 tion seems to be a subject lying nearer to 



one's grasp. The difference between no 

 viscosity and small viscosity — namely, the 

 mathematical assumptions of finite slip — 

 and no finite slip — has already been point- 

 ed out; this requires further investigation. 

 Again, the fact that in all problems 

 hitherto solved, only the first powers of 

 the velocities are taken into consider- 

 ation seems to point to an opening for 

 research. We cannot hope that, even 

 should this single difficulty be overcome to 

 a certain extent, the mechanical difficul- 

 ties would be made much easier. But 

 every step taken ought to lead to some fur- 

 ther insight into the most intractable sub- 

 ject that mathematical physics presents. 

 It must be admitted that the practical en- 

 gineer has almost no use for any of the re- 

 sults obtained by the theory as it at pres- 

 ent stands. It may be answered that pure 

 science does not look to practical ends. 

 This is perfectly true in general. But it 

 can hardly be an argument in the case of 

 hydrodynamics, the very bases of which 

 are assumptions which are supposed to ap- 

 proximate more or less closely to actual 

 conditions. 



In conclusion, I cannot resist making, or 

 rather repeating, an appeal to our pure 

 mathematicians to devote more attention 

 to this subject. The literature is easily 

 accessible. The English treatises of Lamb, 

 Basset, Thomson and Tait will supply 

 most of the needs of a student, while the 

 works of Voight and Kirchhoff and the 

 Handbuch der Physik of Winkelmann may 

 be used to supplement them. Again, 

 Stokes' report of 1846 and that of Hicks 

 in 1881-2 will furnish the main points in 

 the history of the subject. On Vortex 

 Motion we have the later paper of Love in 

 1887 and the presidential address of Hicks 

 to the British Association in 1896. On 

 Tides, Darwin's article in the Encyclopaedia 

 Britannica brings our knowledge up to 

 1888. Most of the work done since 1882 



