Mr. Bjerknes has carried the examination of these phenomena still further in studying experimentally the actions that occur in the depths of the liquid; and for this purpose he has made use of the arrangement shown in Fig. 7. By the side of the vibrating body there is placed a light body mounted on a very flexible spring. This assumes the motion of that portion of the fluid in which it is immersed, and, by the aid of a small pencil, its direction is inscribed upon a plate located above it. By placing this registering apparatus in different directions the entire liquid may be explored. We find by this means figures that are perfectly identical with magnetic phantoms. All the circumstances connected with these can be reproduced, the vibrating sphere giving the phantom of a magnet with its two poles. We may even exhibit the mutual action of two magnets. The figures show with remarkable distinctness--much more distinct, perhaps, than those that are obtained by true magnets. [Illustration: FIG. 7.] However, it must not be thought that these so interesting facts are the result of groping in the dark and the outcome of some fortunate experiment; for they have, on the contrary, been foreseen and predetermined. Mr. Bjerknes is especially a mathematician, and it was a study, through calculation, of the vibratory motion of a body or system of bodies in a medium that led him to the results that he afterwards materialized. After the production, by Mr. Lejeune, of his solutions, Mr. Bjerknes in 1865 entered upon a complete study of the subject, and recognized the fact that the result of such motions was the production of regular mechanical actions. He calculated the directions of these, and, along about 1875, perceived the possibility of reproducing the effects of permanent magnetism. More recently, in 1879, he saw that magnetism by derivation might likewise be explained by those hypotheses, and figured by actions of this kind. It was not till then that he performed the experiments, and submitted a body to the results of calculation. The same process has led him to the conclusion that the action of currents might be represented in the same manner; only, instead of bodies in vibration, it would require bodies in alternating rotation. The effects are much more difficult to ascertain, since it is necessary to employ viscid liquids.