Calculation of Wave Resistance. 516 
If 
F, (y) = | zi (A, cos yu + B, sin yu) du, 
0 
@ (8) 
F, (y) = \ (A, cos yu + B, sin yu) du, 
0 
A, B, being functions of wu, then 
[BO FW) dy =|" AAs + BBs) du, (9) 
assuming that the integrals are convergent. 
To take one of the integrals in (6) as an example, we have 
dr 
ae = — 2y | f (8) e7"? sin (Kgx’ sec 8) cos (kgy sin 6 sec? 6) d@. (10) 
v 0 
To put this into the form (8), we write wu = x, sin 6 sec? 0, then carry out 
the process (9) and finally replace the variable uw in terms of 0; it is clear 
that we shall have to introduce into the integral in the final form a factor 
d@/du; that is, a factor cos* 0/«, (1 + sin? 6). Thus we have 
ele 
“0 br 3 
= 4rK 9c? | a dz if {f (O)}? e207 8ec*? sin? (icga’ sec 0) ae 
5 
= ane? |" (Ff (8)}2 sin? (keqa" sec 6) a (11) 
From (6) we find in this way that the rate of flow of total energy across the 
vertical plane is 
mec? |" { (0)? (3 — sin? 6) sin? cy! se 0) 
Jo (fl ae eB Aenea? exe 0) (12) 
and that the rate at which work is being done across this plane is 
5 
Qnec? i, {Ff (6)}2 sin® (ic x" sec 8) SS (13) 
It is the difference of these two quantities that is significant for our purpose ; 
it is, as would be expected, independent of the time and of the position of the 
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