122 



SCIENCE. 



[N. S. Vol. XVI. No. 395. 



and instability it is necessary to examine 

 the region of convergence of the infinite 

 series which so often present themselves; 

 and this cannot be done with certainty 

 without the methods of function theory. 



In such cases we use the function theory 

 to test the character of the solutions al- 

 ready obtained, and to find out the regions 

 within which they are applicable ; but in 

 the discovery of solutions of new physical 

 problems the methods of general function 

 theory have seldom been used. It is chiefly 

 of its use as an instrument of discovery 

 that I wish to speai to-day. 



It has long been known that the theory 

 of functions of a complex variable is useful 

 in treating the numerous physical problems 

 whose solution can be made to depend on 

 Laplace's equation in two dimensions, 



This equation presents itself in the 

 theory of the two-dimensional potential, 

 and in problems relating to the steady flow 

 of heat, of electricity and of incompressible 

 fluids. 



The essential feature of the method in 

 question is to take an arbitrary function 

 of the complex variable, and to express 

 this function in the form 



f(x + iy)=i,(x, y) +ii'{x, y), 



in which <p and ^' are real functions of two 

 real variables, x and y. 



The functions ?> and 4' are then said to 

 be conjugate to each other, and are in all 

 cases solutions of Laplace's equation, what- 

 ever be the assumed function /'. 



Moreover the two families of curves 



<P {x, y) = C„ 



i' {^, y) = c, 

 (in which Cjand G„ are arbitrary constant 

 parameters) cut each other at right angles. 

 The curves of one system may be taken as 

 equipotential lines, and those of the other 

 system will then be lines of force, or lines 



of flow. The physical boundary of the 

 region must be some one of the lines of 

 either set. 



Some interesting applications of this 

 method to tidal theory have recently been 

 made by Dr. Rollin A. Harris in his 'Man- 

 ual of the Tides,' published by the U. S. 

 Coast and Geodetic Survey.* I would 

 mention especially his use of an elliptic 

 function as the transforming function in 

 the form 



X -\~ iy = sn(if> -\- 1^) . 



The two sets of orthogonal curves drawn 

 by him may be seen in the Annals of 

 Mathematics, Vol. IV., page S3. By im- 

 agining thin walls erected along certain of 

 the stream lines, we see, for instance, the 

 nature of the flow around an island lying 

 between two capes. 



The direct problem of determining a 

 solution of Laplace 's equation that • shall 

 be constant at all points of a boundary 

 previously assigned is usually very diffi- 

 cult. It is a particular case of what is 

 commonly known as the Problem of Dirich- 

 let. Before stating this problem it is con- 

 venient to define a harmonic function. 

 Any real function u{x,y,z) which satisfies 

 Laplace's equation, and which, together 

 with its derivatives of the first two orders, 

 is one-valued and continuous within a cer- 

 tain region, is said to be harmonic within 

 that region. Diriehlet's problem may then 

 be stated as follows : 



To find a function u{x, y, z) which shall 

 be harmonic within an assigned region, T, 

 and which shall take assigned values at 

 points on the boundary surface S. 



This problem has long been one of the 

 meeting grounds of mathematicians and 

 physicists. Some important mathematical 

 theories have received their starting point 

 from this and similar ' boundary- value 

 problems. ' 



* Part IV., A, pp. 574-82. 



