It is my intention to undertake a series of Memoirs on that branch of Hydrodynamics which treats of the transmission of reciprocating motion. It may appear superfluous at first, that any further investigation should be bestowed on the general problem, when it is remembered that two of the most able mathematicians of the present age, M. Poisson and M. Cauchy, have devoted so much of their talents and attention to the subject. Yet, when advances are rapidly making in our experimental knowledge, we generally find that the complex and apparently general results, deduced by those who have paid the greatest attention to the science as it presented itself to their view, are in reality only particular cases, when they have to be applied to experiments in their actual form. This circumstance detracts greatly from the value of almost all general investigations; and in this instance it does so in a remarkable degree. The prosecution of the research has been extended in one direction—the prosecution of experimental knowledge has taken another; and thus, in order to meet the challenge of experimenters to keep pace with them, it behoves the theorist frequently to return on his path, and cross over to the line of their march.