1835.] Philosojjhical Transactions for 1834. 309 



in action at that surface, is very satisfactory when no cause of 

 motion emanates from the fluid itself, and all the forces in action 

 depend merely on the place of a particle, but is defective when ap- 

 plied to fluids consisting of particles that act upon one another by 

 attraction or repulsion ; Clairaut having omitted to attend to the 

 attraction of the stratum, which is not infinitely little in its effect 

 upon the motion of a particle, and is expressed by the difference of 

 two definite integrals. The correction of Clairaut's theory is very 

 important ; because, to him belongs an essential part of the theory of 

 the earth, and he was the first that entertained correct notions 

 respecting the effect to alter the form of the terraqueous globe, pro- 

 duced by heterogeneity in its structure. In the theory of the French 

 philosopher, the equations of the upper surface of the fluid, and of 

 all the level surfaces underneath it, are derived from the single 

 expression of the hydrostatic pressure, and are dependant on the 

 differential equation of the surface. 



They require, therefore, that this latter equation be determinate 

 and explicitly given ; and, accordingly, they are sufficient to solve 

 the problem, where the forces are known algebraical expressions of 

 the co-ordinates of the points of action, but they are not sufficient 

 when the forces are not explicitly given, but depend as they do in 

 the case of a homogeneous planet on the assumed figure of the fluid. 

 In the latter case, the solution of the problem requires farther, that 

 the equations be brought to a determinate form, by eliminating all 

 that varies with the unknown figure of the fluid. 



The author establishes a theory on the subject, applies it to the 

 principal problems of the equilibrium of a homogeneous fluid at 

 liberty, and demonstrates that the figure of equilibrium of a homo- 

 geneous planet can be no other than an oblate elliptical spheroid of 

 revolution. 



Mr. Barlow draws the following inferences from the results of 

 various experiments made to determine the efficiency of paddle-wheels 

 of steam-boats, so constructed as to make the floats enter and leave 

 the water nearly in a vertical position, as compared with common 

 wheels, and with relation to the consumption of fuel: 1. When the 

 wheel is but slightly immersed, little advantage is gained by the 

 vertically acting paddle: 2. When deeply immersed, the vertical 

 paddle has considerable advantage over the common wheel. 



3. When the position of the common wheel is vertical, it affords 

 less resistance to the engine, and is less effective than in any pa t 

 of its revolution, which is exactly reversed in the case of the new 

 wheel. 



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

 power ; because, the velocity of the wheel is not required to exceed 

 that of the vessel in so high a degree, in order to acquire the resist- 

 ance necessary to propel the vessel. 



5. With the same boat and the same wheel no advantage is gained 

 by reducing the paddle so as to bring out, as it is called, the full 

 power of the engine, the effect produced being merely to increase 

 the speed of the wheel, and ^consume steam to no purpose. 



6. With the same boat and the same wheel the speed will be 



