340 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



(21'). 



are 



To find the velocity of transmission we must refer to the equations for the pressure. These 



d^p - - (l^ti - ud^u = (c - m) rf,M, 

 and d„p = - g- ti'y ; 



.•. p = — ^{c - uy — gy — \n°y~ + constant. 



For a particle at the surface p is constant ; and therefore 



constant = {c - li)^ + Zgx + rrx' (23') 



is the equation of the form of a wave : or restoring the value of ?« in terms of x, the 

 equation of the curve of the wave is 



c^A* -x - ct , , 



-— cos'' — — — +9.gx + nz^ = constant (24), 



in which t is supposed constant. 



^Vhen z = h, m = 0; 



.-. constant = c' + 'igk + ri'h' from (23'). 



ch 

 Also when « =^ k, c - u = —; 



k 



ch' 

 .-. constant =—r^ +2gk + n^ k' from (23'); 

 k 



th- 



.: = c= (- - ij - 2g(h -k)- n'{h' - k'); 

 2gli^ 



c= - n" h? = — , 

 h + k 



\h+k) 



ih' I k\-' 



1 cos-'- 



\' \ hi 



.. (25'). 



Before submitting this formula to calculation a few words may be said respecting the 

 experiments of Mr. Russell on negative waves, which without questioning his experimental 

 accuracy in the least degree, I cannot but consider far less satisfactory than those which were 

 made on positive waves. For to generate a perfect solitary negative wave it was necessary that 

 a peculiar law of pressure should have been observed. Unless this law were observed it was a 

 necessary consequence that residuary and superfluous waves would be formed. Now the mode 

 of genesis which Mr. Russell employed seems to have been so little suitable to the nature of the 

 negative wave, that throughout its whole course it seems to have been continually casting off 

 superfluous (or, as Mr. R. calls them, companion) waves. This must have produced a direct 



