42 I'RKSIDENTIAL ADDRESS SECTION A. 



tial c([uations : the details of human experience siippHes the 

 C(~)ns.tants." 



'I^he only details of the develojjnient of the Calculus which 

 J will refer to in connection with Science are : — 



(i.) The Potential. — A great many of the vectors (or 

 directed quantities)— forces, velocities, etc. — that crop up so 

 inevitabl\- in mechanical problems, were noticed to have one 

 strikin*;' (juality — the three components in space were the rates 

 of change in their directions of some mathematical expression 

 or function ; as the flow of heat in any direction depends on the 

 change in the direction of the ])oint of the stun of the mass of 

 every element of the attracting body divided b\- its distance 

 from the point. In general, the force exerted by any system 

 in an}- direction is the " gradient " or rate of change in that 

 direction of the potential energy. This quantity was happily 

 named by Green (a self-educated Nottingham miller) in 1828 

 the " potential " of the system. In every branch of Science it is 

 our most effective wea]ion of attack, and the equation 



^ ^ V ^ ^ \^ ^ ^ V 



K — r, 4- 7r-7, + K — 7i — 4 -rrp. connecting this ix>tential (V) 

 b x- y o z-' 



with the density (p) of matter electricity or heat, seems often 

 to sum up the position completely. This equation is due to 

 Laplace and Poisson. 



(ii.) There are three purely mathematical propositions 

 known by the names of their discoverers, Stokes, (iauss, and 

 Green. Stokes' connects the value of a quantity round any closed 

 circuit with its value over any caf> having the circuit for boundary. 

 That of Gauss connects in the same way the values over a closed 

 surface with the values throughout the interior, and Green's 

 gives the value at any outside }X3int in terms of the values on 

 the surface and inside. These theorems make many experi- 

 mental results of apparent complexity mere logical consequences 

 of simpler facts. Though fairly complex in their origin, and far 

 removed from the intuitions of common sense, they are practi- 

 cally properties of geometrical space. 



One cannot omit among the debts of science to mathematics 

 a theorem established by P'ourier in connection with the 

 theory of heat. From what is essentially nothing more 

 than an identity in elementary trigonometry. Fourier 

 evolved simply a formula, whose physical meaning is, that any 

 oscillatory or periodic motion consists of a series of simple wa^•e 

 motions whose periods are the submultiples of the main 

 period. In short, it extends the idea of " harmonics," .so familiar 

 to musicians, to all cases of j^eriodic motion. The well-known 

 principles of physical reasoning known as Conser^'ation of Angu- 

 lar Momentum, Conservation of Energy, Principle of Least 

 Action, D'Alembert's Princi])le, are only convenient summaries 

 of Newton's Laws, each, as it were, a sort of railhead in which 

 a host of results are summarized, and from which attacks on 

 new ])n)blems can -tart with cct.nomy of time and thioufjht. But, 



