RECENT ADVANCES IN SCIENCE 183 



hand, and Applied Mathematics and Astronomy on the other. 

 There is further the additional difficulty that many problems 

 in Pure Mathematics are obviously and directly suggested by 

 some natural phenomenon when formulated in symbolical 

 notation. All investigations on the solution of Laplace's 

 equation and its modifications, as well as the other equations 

 of mathematical physics, come under this category, and it is 

 no easy task to decide whether the pure or the applied 

 mathematician may claim them as coming within his sphere. 



In this survey of some of the recent work in applied mathe- 

 matics the writer's principle has been to limit himself to the 

 consideration of such papers as can be definitely excluded from 

 the domains of pure mathematics, astronomy, and mathe- 

 matical physics. We shall then report on the progress in the 

 development of Statics and Dynamics of Solids and Fluids, 

 including the Theory of Gravitational Attraction, Motion in a 

 Resisting Medium, and the Mechanics of Astronomy considered 

 from the mathematical point of view. We omit Relativity 

 as such, since it is of more immediate interest to the physicist 

 and the astronomer. 



Research in any branch of science can be divided into two 

 categories. The first consists of the further development of 

 what may be called the classical problems of the subject, the 

 working out of the further consequences of the well-established 

 principles and methods, and the elaboration of the detailed 

 solutions. The second category of research concerns itself 

 with the discovery of new principles, the development of new 

 methods, and the attacking of new problems suggested by 

 theoretical, practical, and industrial applications. In applied 

 mathematics this dual nature of research is at once made clear 

 by the recent growth of the subject. W T hereas on the one 

 hand we have much that is a continuation or an extension of 

 the well-established ideas and processes, we have on the other 

 hand ample evidence of progress towards fresh fields of in- 

 vestigation. 



Treating first of the research of a pioneer nature we are con- 

 fronted by two main themes. The principle of Relativity has 

 necessitated a reconsideration of the fundamental principles of 

 mechanics, and this has brought in its train a certain amount 

 of scepticism as to the accepted facts of universal gravitation. 

 It is assumed in ordinary dynamics that the gravitational effect 



