542 Lord Rayleigh on Convection Currents in 



squares. If the sides of the squares be 2ir, we may take 

 with axes parallel to the sides and origin at centre 



w = cos x + cos y, 



(55) 



being thus composed by superposition of two parts for each 

 of which & 2 = 1. This makes dwjdx = — sin x, vanishing 

 when #=+7]-. Similarly, dwjdy vanishes when y=±7r, so 

 that the sides of the square behave as fixed walls. To find 

 the places where w changes sign, we write it in the form 



x -\-y x — y 

 w = 2 cos '- . cos — j~ , 



. . (56) 



giving x+y—±ir, x—y—±ir, lines which constitute the 

 inscribed square (fig. 1). Within this square w has one sign 

 (say -f ) and in the four right- 

 angled triangles left over the 

 — sign. When the whole plane 

 is considered, there is no want 

 of symmetry between the + and 

 the — regions. 



The principle is the same 

 when the elementary cells are _ 

 equilateral triangles or hexa- 

 gons ; but I am not aware that 

 an analytical solution has been 

 obtained for these cases. An 

 experimental determination of 

 k 2 might be made by observing 

 the time of vibration under gravity of water contained 

 in a trough with vertical sides and of corresponding- 

 section, which depends upon the same differential equation 

 and boundary conditions *. The particular vibration in 

 question is not the slowest possible, but that where there 

 is a simultaneous rise at the centre and fall at the walls all 

 round, with but one curve of zero elevation between. 



In the case of the hexagon, we may regard it as deviating 

 comparatively little from the circular form and employ the 

 approximate methods then applicable. By an argument 

 analogous to that formerly developed f for the boundary 

 condition w = 0, we may convince ourselves that the value 

 of k 2 for the hexagon cannot differ much from that appro- 

 priate to a circle of the same area. Thus if a be the radius 



* See Phil. Mag. vol. i. p. 257 (1876); Scientific Papers, vol. i. 

 pp. 265,271. 

 t Theory of Sound, § 209 ; compare also § 317. See Appendix. 



