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surface of contact the nature of steel; there will be a considerable permanent magnetism, and the transient magnetic force, which alone produces the movement, will be weakened in proportion. A number of experiments which I have made upon the magnetic force of a bar of soft iron bent into a horseshoe (of which I shall speak hereafter) has shown me the great disadvantages of oft-repeated shocks, proceeding from the sudden contact of the armature. But, if we stop at the mechanical principles of magnetism, it may be objected that the active force gained will not be absolutely lost for the purposes of utility; that in part the elasticity of the iron will itself reproduce it; that another portion may be regained by springs properly applied, or by other mechanical methods which may be invented. We leave the appreciation of all these factitious means and of these superadditions to those who are in the habit of constructing machines; they well know their insufficiency, the great loss of working power, and how rapidly all the systems are destroyed, unless the greatest care be paid to the preservation of the active forces. But we must seek the means of this preservation in the nature of the forces themselves. The history of the steam-engine teaches us that its improvement commences with Watt's ingenious idea of stopping the escape of the steam before the piston had accomplished its stroke, and of causing the steam afterwards to act by its own expansion. Watt understood the subject: all he did was to give to the function $P=\phi (s)$, which expresses the action of the steam, such a form as $\int _{0}^{q}P\,ds=\int _{0}^{a'}P'ds'$, and thus the active force gained becomes zero, all the prejudicial and destructive vibrations in the machines previously constructed cease, and the power of the motive force is converted for the most part into useful action. I must here cite the valuable researches of M. Poncélet on the construction of hydraulic wheels,—a work founded upon a profound comprehension of the same principles.

These considerations, at once clear and simple, have induced me to reject entirely every apparatus in which magnetism is applied to produce immediately an oscillating motion; these constructions being, as we have seen, as inadmissible as they are impracticable of execution on a large scale.

3.

In the note which I had the honour of laying before the Academy of Sciences of Paris I stated that, in accordance with all experiments, magnetism is a power acting like universal gravitation, solely in some function of space. The integral ${\frac {\int _{0}^{a}M\,ds}{a}}$ comparable with the known number $g$, represents the mean action furnished by the attraction of two