286 BELL SYSTEM TECHNICAL JOURNAL 



from its original contour till it becomes unrecognizable. Definite 

 as the wave-length and the wave-speed and the waves themselves 

 may seem at times to be, at other times they seem indefinite and 

 indefinable. 



Now in the general theory these difficulties are removed, for the 

 attention is focussed first of all upon an abstract entity with a neutral 

 name — the "phase." There is a differential equation, a "wave- 

 equation," governing the phase; and this entity is propagated in a 

 certain very definite way conforming to the vague description which 

 I gave above, and it has a wave-length and a wave-speed. As for the 

 elevations and depressions of the water-surface, they copy the varia- 

 tions of the phase more or less faithfully, and may be computed from 

 these; but except in particular cases the copy is not exact. To the 

 theorist, the ripples upon the water appear as the secondary and 

 imperfect manifestations of an abstract wave-motion, discernible only 

 to the eye of the mind. 



That the undulatory theory should introduce an abstraction even 

 into the example whence it sprang, requiring us to imagine waves of 

 phase underlying the tangible waves of the sea, is not at all remarkable. 

 Often in physics a theory evolves in this way. It begins when some 

 one notes a resemblance between two or more phenomena; it continues 

 by the invention of a neutral and colorless mathematical expression 

 for describing the common aspect of all these phenomena; and then 

 the theorists take over the mathematical expression and transform 

 and generalize and extend it, until the theory, which at first was 

 a casual statement that two different things are in some ways much 

 alike, eventually is defined as the entire system of solutions of some 

 differential equation. At present there seems to be no adequate way 

 of defining the term "wave-theory," except to say that wherever a 

 certain differential equation is introduced and solved, there a wave- 

 theory is adopted. Moreover, it is open to anyone to adduce new 

 differential equations more or less like the first one, and define as a 

 wave-theory any which involves the solutions of these. Then a wave- 

 motion is any motion which conforms to one of the solutions; and 

 when it comes to defining a wave, what can anyone say except that 

 under certain restricted conditions a wave-motion may resemble a 

 procession of ripples on water? 



Such a consummation may be devoutly wished by the mathemati- 

 cian ; to the physicist and to the expositor it is not always so welcome. 

 As a theory increases in scope through increase of abstraction, it loses 

 the picturesqueness which for many minds is its reason for being. 

 One climbs and climbs, and the view indeed grows wider, but the 



