Free Procession of Waves in Deep Water. 407 
Hence, for the harmonic expansion (24) of aid 0), we have 
= wet Zi. 2rex al 2m 
‘i a au ae Yas wea ae ae 
: (33). 
The imaginary form of the last member of this equation 
facilitates the evaluation of the integral. Instead of cos eu 
in the last factor, substitute 
2ra 
am ay 
277 
cos J ave 6sin or € 
. a (34). 
The alternative makes no difference in the summation 
+o 
{ dx, because the sine term disappears for the same reason 
that the sine terms in (29) disappear because of (30). Thus 
(33) becomes 
A=>{rst| de a ae age): 
put now ,/(z+ ur) = 
dx 
whence Uwe 2do, and tw=—a?—z . . (86). 
Using these in (385) we may omit the instruction {RS} 
because nothing imaginary remains in the formula: thus 
we find 
@) oY xe 277, 2njz a 
| adage r ge ey ee vee es ON, (Se 
J 
ao) 
The transition in (37) is made in virtue of Laplace’s cele- 
brated discovery Ns doe, = / * 
§ 17. Equation (38) allows us eran to see how near to a 
curve of sines is the graph of P(#, 0), for any particular 
value of A/z. It shows that 
~ 2rz Anz dre 
7 
A,=—2e %; As/Aj=,/h.€ 2; AsfAs=4/2.€ *5 (38). 
