STABILITY; PROPULSION, AND SEA-GOING QUALITIES OF SHIPS. 19 



it therefore becomes necessary to find some auxiliary formula. In the 

 ' Philosophical Transactions ' for 1863, pp 134-37, Mr. Rankine has shown that 

 the augmented surface of a trochoidal ribbon on a given base and of given 

 breadth may be found by multiplying their product by the following coefii- 

 cient of augmentation : — 



l+4(sin0)H(sin0)\ 



in which is the angle which the inflexional tangent makes with the base. 

 For a ship in which the stream-lines or tracks of the particles of water are 

 trochoids, it would be a sufficient approximation to integrate, 



length X I breadth x {1 +4 (sin ^)^ + (sin ^y} 



with regard to the draught of water, considering both the angle (p and the 

 half-breadths as variable elements to be determined from the drawings. 

 AVhere the stream-lines are not trochoids, f may be taken as the angle of 

 greatest obliquity. But the theory has been only partially extended to three 

 dimensions ; and indeed if it were possible to do so, the mere introduction 

 of a third variable would not meet the case, unless account were taken of 

 the vertical displacement of the surface of the water consequent upon the 

 uniformity of pressure at that surface. 



The resistance determined by the calculation of the augmented surface 

 includes in one quantity both the direct adhesive action of the water on the 

 ship's skin, and the indirect action through increase of the pressure at the 

 bow and diminution of the pressure at the stern. 



For the coefficient of friction, Professor Eankine takes /=0-0036 for sur- 

 faces of clean painted iron. This is the constant part of the expression 

 deduced by Professor Weisbach from experiments on the flow of water in pipes. 

 The corresponding coefficient deduced from Darcy's experiments is 0-004. 



The augmented surface in square feet, multiplied by the cube of the speed 

 in knots, and divided by the I. H. P., gives Hanline's coefficient of propulsion* . 

 In good clean iron vessels this ranges about 20,000 ; while in H.M. Yacht 

 'Victoria and Albert' (copper sheathed) it reached 21,800. Its falling 

 much below 20,000 is considered to indicate that there is some fault cither 

 in the ship or in her engines or propeller, or else that the vessel is driven 

 at a speed for which she is not adapted. 



Professor Eankine adds that " as for misshapen and Hi-proportioned ves- 

 sels, there does not exist any theory capable of giving their resistance by 

 previous computation." 



This, again, raises the question. What are good forms ? According to Pro- 

 fessor Eankine's theory, they are forms along which a particle of water can 

 ghde smoothly. Among these, as a particular case, Mr. Scott EusseU's wave- 

 lines appear to be included. But these are by no means the only ones which 

 satisfy the problem of smooth gliding, or of stream-lines. Another method 

 of constructing curves fulfilling this condition has been given by Mr. Raukine 

 in a series of papers published respectively in the ' Philosophical Transac- 

 tions ' for 1863, p. 369, and in the ' Philosophical Magazine ' for October 

 1864, and January 1865. Elementary descriptions of this method are given 

 in the 'Engineer' of the 16th of October 1868, and in a treatise on ' Ship- 

 building : Theoretical and Practical.' Their theory has not yet been carried 

 very far; and when we have reference to three dimensions, it does not appear 



* For examples of that coeiRcient, see the ' Civil Engineer and Architects' Journal ' for 

 October 1860, and the " Eeport of the Committee of the British Association on Steamship 

 Performances," 1868. 



c2 



