454 Royal Society. 



velocities arising from this form of the apparatus ; and gives an ac- 

 count of the results of various experiments made on its efficiency as 

 compared with the common wheels, and with relation to the con- 

 sumption of fuel. The general results to which he is led are as fol- 

 low : — 1st. When vessels are so laden as that the wheel is but 

 slightly immersed, little advantage is derived from the vertically acting 

 paddles. 2ndly. In cases of deep immersion, the latter has consi- 

 derable advantage over the wheel of the usual construction. 3rdly. In 

 the common wheel, while the paddle passes through the lower portion 

 of the arc, that is when its position is vertical, it not only affords 

 less resistance to the engine, but is less effective in propelling the 

 vessel than in any part of its revolution. 4thly. The paddle of the 

 wheel, while passing through the lower portion of the arc, affords 

 more resistance to the engine, and is more effective in propelling the 

 vessel, than in any part of its revolution j a property which is a serious 

 deduction from its value j for, in consequence of the total resistance 

 to all the paddles being so much less than in the common wheel, 

 much greater velocity is required to obtain the requisite pressure, 

 and a greater expenditure of steam power is incurred. This loss of 

 power is most sensible when the wheel is slightly immersed j but in 

 cases of deep immersion the vertical paddle has greatly the advantage, 

 othly. In any wheel, the larger the paddles the less is the loss of* 

 force ; because the velocity of the wheel is not required to exceed that 

 of the vessel in so great a degree, in order to acquire the resistance 

 necessary to propel the vessel, fithly. With the same boat and the 

 same wheel no advantage is gained by reducing the paddle so as to 

 bring out the full power of the engine ; the effect produced being sim- 

 ply that of increasing the speed of the wheel, and not that of the ves- 

 sel. 7thly. An increase of speed will be obtained by reducing the 

 diameter of the wheel ; at least within such limits as allow of the 

 floats remaining sufficiently immersed in the water; and provided the 

 velocity of the engine does not exceed that at which it can perform 

 its work properly. Sthly. An advantage would be gained by giving 

 to the wheel a larger diameter, as far as the immersion of the pad- 

 dles produced by loading the vessel would not so sensibly affect the 

 angle of inclination of the paddle ; but this advantage cannot be ob- 

 tained with an engine of the same length of stroke, because in order 

 to allow the engine to make its full number of strokes, it will then be 

 necessary to diminish the size of the paddles, which is a much 

 greater evil than having a wheel of smaller diameter with larger 

 paddles. 



The reading of a paper was then commenced, entitled, " On the 

 Equilibrium of a Mass of Homogeneous Fluid at liberty." By James 

 Ivory, Esq., K.H., M.A., F.R.S. 



June 5. — Mr. Ivory's paper, entitled, " On the Equilibrium of a 

 Mass of Homogeneous Fluid at liberty," was resumed and concluded. 



The author shows that Clairaut's theory of the equilibrium of fluids, 

 however seductive by its conciseness and neatness, and by the skill 

 displayed in its analytical construction, is yet insufficient to solve 

 the problem in all its generality. The equations of the upper surface 



