Deep Water Ship-Waves. 747 



between and C. Imagine the curve to be exactly sinu- 

 soidal throughout, and continued sinusoi dally to cut the zero 

 line at C C. 



We should thus have in C C a train of 4 J- sinusoidal waves; 

 and if the same is continued throughout the infinite proces- 

 sion .... C C C .... we have a discontinuous periodic curve 

 made up of continuous portions each 4| periods of sinusoidal 

 curve beginning and ending with zero. The change at each 

 point of discontinuity C is merely a half-period change of 

 phase. A slight alteration of this discontinuous curve within 

 b'0° on each side of each C, converts it into the continuous 

 wavy curve of fig. 15, which represents the water-surface due to 

 motion at speed V ' ga^ir of the pressural forcive represented 

 by the other continuous curve of fig. 15. 



§ 49. Every word of § 48 is applicable to figs. 14 and 13 

 except references to speed of the forcive, which is ^/gajl^ir 

 for fig. 14 and ^/gaj^lir for fig. 13; and other statements 

 requiring modification as follows : — 



For 4^ " periods " or " waves," in respect to fig. 15 ; sub- 

 stitute 9 } 2 in respect to fig. 14, and 20^ in respect to fig. 13. 



For " depression " in defining MM in respect to figs. 15, 

 14 ; substitute elevation in the case of fig. 13. 



§ 50. How do we know that, as said in § 48, the formula 

 {(83), (86), (87)} gives for a wide range of about 120° on 

 each side of (9=180°, 



d(^)=(-iyd(180°).sin0>^ . . . (88), 



which is merely § § 48, 49 in symbols ? it being understood 

 that j is any integer not <4 ; and that e is *9, or any numeric 

 between *9 and 1 ? I wish I could give a short answer to this 

 question without help of hydrokinetic ideas ! Here is the 

 only answer I can give at present. 



§ 51. Look at figs. 12-16, and see how, in the forcive de- 

 fined by e = '9, the pressure is almost wholly confined to the 

 spaces 6<60° on each side of each of its maximums, and is 

 very nearly null from = 60° to = 300°. It is obvious that 

 if the pressure were perfectly annulled in these last- 

 mentioned spaces, while in the spaces within 60° on each side 

 of each maximum the pressure is that expressed by (74), 

 the resulting motion would be sensibly the same as if the 

 pressure were throughout the whole space CC (6 = ()° to 

 6 = 3(50 C J, exactly that given by ("74 ). Hence we must expect 

 to find through nearly the whole space of 240°, from 60° to 

 300°, an almost exactly sinusoidal displacement of water- 



