REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 135 



motion of a single particle, in the place of the motion of an 

 aggregate of particles. Though this mode of deriving them is 

 the best possible on account of its simplicity, it was yet de- 

 sirable to know how they may be obtained from the general 

 equations of fluid motion. In a paper contained in the Trans- 

 actions of the Philosophical Society of Cambridge *, the author 

 of this Report has given a method of doing this, both for incom- 

 pressible and elastic fluids, and has shown that a term in the 

 general formulae which gives rise to the complexity common to 

 most hydrodynamical questions, disappears for this kind of 

 motion. Euler had already done the same for incompressible 

 fluids f . The instances in nature of fluid motion of the steady 

 kind are far from uncommon ; and it is probable that when the 

 equations applicable to them are better known, and studied 

 longer, they may be employed in very interesting researches. 

 The motion of the atmosphere, as affected by the rotation of 

 the earth, and a given distribution of the temperature due to 

 solar heat, seems to be an instance of this kind. 



We will now proceed to consider in order the principal hydro- 

 dynamical problems that have recently engaged the attention 

 of mathematicians. For convenience we shall class them as 

 follows : — I. Motion in pipes and vessels. II. The velocity of 

 propagation in elastic fluids. III. Musical vibrations in tubes. 

 IV. Waves at the surface of water. V. The resistance to the 

 motion of a ball-pendulum. 



I. The motion of fluids in pipes and vessels has not been 

 treated with any success, except in the cases in which the con- 

 dition of steadiness is fulfilled. The paper above alluded to, 

 in the Transactions of the Philosophical Society of Catnbridge, 

 contains some applications of the equation of steady motion for 

 incompressible fluids, to determine the velocity of water issuing 

 from different kinds of adjutages in vessels of any shape : also 

 a theoretical explanation of a phsenomenon which a short while 

 ago excited some attention, — that of the attraction of a disc to 

 an orifice through which a steady current either of water or air 

 is issuing. 



In the Memoirs of the Paris Academy of Sciences X there is 

 an Essay by M. Navier on the motion of elastic fluids in ves- 

 sels, and through different kinds of adjutages into the sur- 

 rounding air, or from one vessel into another. For the sake of 

 simplicity the author considers the fluid to be subject to a con- 

 stant pressure, and consequently the motion to have arrived at 

 a state of permanence. His calculations are founded upon the 



• Vol. iii. Part III. f Memoires de I'/lcadimie de Berlin, 1755, p. 344. 

 X Tom. ix. 1830. 



