On the Magic Square of the Knight's March. 101 



the integral of which is V = ty(s—a l t). In the same way it may 

 be shown that the density (p) must be a function of s — a x t in 

 the case of a constant rate of propagation. 



Now these criteria of uniform propagation are not satisfied 

 by the hypothesis of plane waves, because on that hypothesis 



we have V=a Nap. logp=/j>-(* + V)/). 



Still less are they satisfied by the hypothesis of spherical waves. 

 But they are fully satisfied on the hypothesis of non-diver- 

 gent waves, because the above value of <p shows that in this 



case -5- is a function of z— a't; and from the known general 



equation 0XT , dp , V 2 



1 a 2 Nap>]ogp+ ^ + _ ==0j 



it follows that p is also a function of z — a't. 



I have thus shown, by reasoning with exact equations, the 

 entire compatibility of the hypothesis of non-divergent waves 

 with the hydrodynamical equations. Having at the same time 

 demonstrated the incompatibility of plane- waves and spherical 

 waves, I consider that the theorems of capital importance, to 

 which the reasoning in this and the two former communica- 

 tions has been directed, are established ; viz. that non-diver- 

 gence is the normal character of aerial waves, and that the 

 velocity of propagation is greater than a. I propose in a fu- 

 ture communication to draw some inferences from the equa- 

 tions (B.) and (C.) 



Cambridge Observatory, 

 July 20, 1848. 



XVIII. On the Magic Square of the Knight's March. 

 By William Beverley. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



I INCLOSE for insertion in the Philosophical Magazine 

 a very interesting Magic Square, formed by numbering 

 consecutively the moves of the knight in the grand tour of the 

 chess-board. The knight's march has engaged the ingenuity 

 of many eminent philosophers and mathematicians; but I 

 believe that Mr. W. Beverley is the first who has solved the 

 difficult problem of converting it into a magic square. The 

 principle upon which he has effected it, seems to be somewhat 

 akin to that invented by Dr. Roget, S.R.S., as explained in 

 his paper on the Knight's Move in vol. xvi. of the Philoso- 

 phical Magazine. 



Yours very faithfully, 

 5 Smith Street, Chelsea, H. Perigal, Jun. 



March 29, 1848. 



