514 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



and by substituting a^a^c^ for b% and transposing it, becomes 



2'^ o 



(,«*+ -"■'') (l-.^. a '(e"'^- e-^'^f \ = ^ (e"" - e''^') 



2__ art — a h 



TT „ e — e 



c^ - 9 



the same result as in art. 10. 

 The second result is 



«^ ' ^_Z! ■,.«(,«/' _ ,-«/')2 + ^ ... ..^./' , .-«& 



_o_. ^^^^ .^^^^^., _ -.ky ^ 2Zi 6. (,^./' + e-"-') 



— ^ be - g—— (e^«'' - e '*«'') 



Whence « = j^^^^ \ —-{He'^' - ^-^^") - ^ ^^ («"'' - .-">' + -^ .^ I 



c^ (e""- + e 



<*■ ^ o — *■- ''\3 ,.« /* „ — K A 





„ /a. II „ — a ft \ 







These two results give us, respectively, the velocity of transmission and the 

 height of the wave. 



If h be small compared with A, the above equations give 



(^ =gh 



A 



« = 7^ — 



as an approximation. 



The first result is too well known to require comment ; the second, if it have 

 any truth at all, appears to shew that the tendency of waves in shallow water is 

 to become semicircular, measuring from the mean points to the crests. 



12. Before we proceed further, it will be convenient to make a trifiing al- 

 teration, both in the mode of proceeding and in the notation. 



Let now represent x—et instead oi ct—x, and denote u by the sum of a 

 series of terms of the form 



bo + b, (e'^y + e-'^y) sin + &c. 



which is the most general form of which it is susceptible for a wave motion. 



