754 Lord Kelvin on 



dW=(2y + iy[-^« og J-±^ + i + ?l 1 + ^ + ... + |^ 



#(180°) = ( - \y («,■ + iy[j.*tanTi 2^f + ! _ ^_! + c 2 + . . . 



+ (-l)'-'^M-l)^] (93). 



The asymptote of d(0°) shown in the diagram is explained 

 by remarking that when j is infinitely great, the travelling 

 velocity o£ the forcive is infinitely small ; and therefore, by 

 end of § 41, the depression is that hydrostatically due to the 

 forcive pressure. This, at 6 = 0°, is equal to 



A + e 1-9 QK 



1 — e z 



§ 63. Tlie interpretation of the curves of fig. 17 for points 

 between those coi responding to integral values of j is ex- 

 ceedingly interesting. We shall be led by it into an investi- 

 gation of the disturbance produced by the motion of a single 

 forcive, expressed by 



n=J^- 2 (94); 



but this must be left for a future communication, when it will 

 be taken up as a preliminary to sea ship-waves. 



§ 64. The plan of solving by aid of periodic functions the 

 two-dimensional ship-wave problem for infinitely deep water, 

 adopted in the present communication, was given in Part IV. 

 of a series of papers on Stationary Waves in Flowing Water, 

 published in the Philosophical Magazine, October 1856 to 

 January 1887, with analytical methods suited for water of 

 finite depths. The annulment of sinusoidal waves in front 

 of the source of disturbance (a bar across the bottom of the 

 canal), by the superposition of a train of free sinusoidal waves 

 which double the sinusoidal waves in the rear, was illustrated 

 (December 1886) by a diagram on a scale too small to show 

 the residua] disturbance of the water in front, described in 

 § 53 above, and represented in figs. 18, 19, 20. 



In conclusion, I desire to thank Mr. J. de Graaff Hunter 

 for his interested and zealous co- operation w T ith me in all the 

 work of the present communication, and for the great labour 

 he has given in the calculation of results, and their represen- 

 tation by diagrams. 



