MancJiester Memoirs, Vol. I. ( 1 906), No. 8. 1 7 



compared with the cumbrous expressions from which they 

 have been derived, and also the rapid fall in the leading- 

 coefficients in the implicit equation to the wave-form 

 contained in (25) and (26) after the first trigonometrical 

 term, and the close approximation of that wave-form to 

 a trochoid. 



The equations also indicate that for a closer know- 

 ledge of the form of finite waves, we need more terms in 

 the coefficients of the earlier trigonometrical terms rather 

 than a greater number of terms. On this account no 

 diagrams of the wave profile have been drawn. 



The Wave-Profile. 



Returning to equations (25) and (26), which give the 

 implicit equations of the wave-profile, we will now write 



?-• 



and 



so that 



^=i^/^'- + 4/^* + ^/A 

 \ 4 



Then the equations may be written 

 x = ^ (ct + a) 



Kg 



-—(h + ^//' + —Jf ) sin ^{ct + a) 



2iv\ 2 4 / Aq 



27r\3 18 y Ao 



A 15 ,, . 67r, 



27r 72 A/ ' 



27r 00 A„ 



