Proceedings of Philosophical Societies, [April, 



to be renewed. But a very important difference, and one which 

 will be favourably received by the public, is the diminution of the 

 price. From all these circumstances, we think it right to infer, 

 that the lamp deserves the approbation of the Academy. 



A Treatise upon Wheels for raising Water ; by M. Navier. — 

 Committee, Messrs. de Prony, Fourier, and Dupin, Secretary. 



M. Navier undertakes to determine the proportion between the 

 vis motrix and the effect produced in rotatory machines for 

 raising water. 



The principle of the preservation of the vis viva gives a mathe- 

 matical relation between the four species of forces that remain 

 to be considered in the problem, if we neglect the friction and 

 the cohesion of the water, which indeed are very little. This 

 principle, which was discovered by Huygens, was placed by 

 John Bernoulli in the number of the fundamental laws of dyna- 

 mics; Daniel made some fortunate applications of it; and Borda 

 used it with great success in the calculations of several machines 

 of which water was the moving power. In those which M. 

 Navier notices, it is, on the contrary, the water which is raised 

 by some other foreign power. We owe to Borda the first exact 

 calculation of the vis viva lost, but he only gave it for particular 

 cases. To M. Carnot we are indebted for the general law which 

 he has clothed in the following theorem. " In every case of a 

 body in motion, passing from one situation to another, the sum 

 of the quantities of action which have been during this interval 

 impressed by all the forces is always numeiically equal to half 

 the sum of the vis viva acquired at the same time by the differ- 

 ent bodies in question, plus the half of the vis viva lost by the 

 effect of the sudden changes of velocity, if there have been such 

 changes." 



Wheels for raising water are divided into three classes, 

 according as the rotatory axis is horizontal, vertical, or inclined. 



In the bucket wheel there is a vis viva acquired by the water 

 at the instant one of the buckets is filled, and another lost at the 

 instant it is emptied. From the above-mentioned law may be 

 gained the proportion of the vis motrix to the effect of the 

 machine ; and by a simple differentiation, we obtain the velocity 

 that gives the most favourable proportion. 



In the drum wheel there is not any power lost ; hence this 

 wheel is more advantageous than the preceding one. 



M. Navier describes very minutely the spiral pump formed by 

 a tube of an uniform or variable size, bent in a spiral on a cone 

 whose axis is horizontal. This ingenious machine has the very 

 important merit of producing more beneficial effects in proportion 

 as it is used in raising water to a greater height. A calculation 

 of M. Navier fixes the height at which that property begins to 

 be very perceptible. 



If we fasten to a vertical axis a syphon inchned in such a 

 manner as to rise in a contrary direction to the rotary motion, 



