a Horizontal Layer of Fluid. 541 



In a particular solution where w as a function of x and y 

 has the simplest form, say 



w = 2 cos x . cos y, (51) 



the particular coefficients of # and ?/ which enter have 

 relation to the particular axes o£ reference employed. If 

 we rotate these axes through an angle </>, we have 



w = 2 cos {x' co$<p — y' sin<£} . cos -jx' sin 4>-\-y' cos c/>} 

 = cos {#'(cos <p — sin </>) } . cos {?/' (cos </> + sin <j>) } 

 + sin {^'(cos <f> — sin cj>) \ . sin {?/(cos <f> 4- sin 0) } 

 -f cos \x' (cos</>+ sm<£)|- . cos {?/(cos <£ — sin (£)} 

 — sin {#'(cos <£ + sin (/>)} . sin {y (cos </> — sin (/>)}. (52) 

 For example, if </) = J7T, (52) becomes 



w = cos (/ v/2) + cos (* V 2 )- • • • (53) 



It is to be observed that with the general value of (f>, if 

 we call the coefficients of x\ y', I and m respectively, we 

 have in every part / 2 +??i 2 = 2, unaltered from the original 

 value in (51). 



The character of w, under the condition that all the 

 elementary terms of which it is composed are subject to 

 / 2 + m 2 = constant (F), is the same as for the transverse dis- 

 placement of an infinite stretched membrane, vibrating with 

 one definite frequency. The limitation upon w is, in fact 

 merely that it satisfies 



{d 2 /dx 2 + d 2 /dy* + k 2 )w = (54) 



The character of w in particular solutions of the mem- 

 brane problem is naturally associated with the nodal system 

 (w — 0), where the membrane may be regarded as held fast ; 

 and we may suppose the nodal system to divide the plane 

 into similar parts or cells, such as squares, equilateral 

 triangles, or regular hexagons. But in the present problem 

 it is perhaps more appropriate to consider divisions of the 

 plane with respect to which w is symmetrical, so that dw/dn 

 is zero on the straight lines forming the divisions of the 

 cells. The more natural analogy is then with the two- 

 dimensional vibration of air, where w represents velocity- 

 potential and the divisions may be regarded as fixed walls. 



The simplest case is, of course, that in which the cells are 



