On the Development of Mathematical Analysis and Its Relations to some other Sciences

Without restricting ourselves to the historical order, let us resume the development of mathematical physics in the last century, in so far as analysis is concerned. The problems of thermal equilibrium lead to the equation already known to Laplace in the study of attraction. There are few equations which have been the object of so many researches as this celebrated one. The conditions for the limits may assume various forms. The simplest case is that of the thermal equilibrium of a body, the elements of the surface of which are maintained at given temperatures. From the physical point of view, it may be regarded as evident that the temperature, assumed continuous in the interior since there is no source of heat, is determined when it is given at the surface. The more general case is that in which, the condition remaining permanent, there would be a radiation outward with an intensity • varying at the surface according to a given law ; in particular the temperature may be given over one portion, while there is radiation over the remainder. These questions, which are not yet solved in their widest generality, have largely contributed to the direction taken by the theory of partial differential equations. They have called attention to types of determination of the integrals which would never have presented themselves if we had been restricted to a purely abstract point of view. Laplace’s equation has been already met with in hydrodynamics, and in the study of attraction varying inversely as the square of the distance. The latter theory brought to light elements of the most essential nature, such as the potential of single and double layers. Here we meet with analytical combinations of the highest importance, which have since been notably generalised. Green’s formula is a case in point. The fundamental problems of electrostatics are of the same order of ideas, and certainly the celebrated theorem on electrical phenomena in the interior of a hollow conductor, which Faraday rediscovered at a later stage by experimental means, knowing nothing whatever of Green’s memoir, was a notable triumph for theory. This magnificent aggregate has remained the type of the classical theories of mathematical physics, which seem to us to have almost attained perfection, and which have exercised, and still exercise, so happy an influence on the progress of pure analysis by suggesting to it the most beautiful problems. The theory of functions again will afford us a notable comparison. The analytical transformations brought into play are not distinct from those we have met with in the steady movement of heat. Certain fundamental problems in the theory of functions of a complex variable have lost their abstract enunciation and assumed a physical form, as in the case of the distribution of temperature on a closed surface of any connectivity whatever and without radiation, in thermal equilibrium, with two sources of heat which necessarily correspond to equal and opposite flows. Interpreting this, we find a question on Abelian integrals of the third species in the theory of algebraical curves.