496 Dr. T. H. Havelock. 
There is also a similar expression corresponding to the pole a,+7b,; from 
(21) we see that the result can be written in the form 
X = (Ai + 7By) #407" —( Ay + 1By) etre nbo, 
Hence for the resistance we have, from (7), 
gpR/K? — (Ay?+ B;?) e72hk + (Ap?+ B,?) E72box 
—2 {(A1 Az +B, Bz) cos 2% —(A2B,— A,Bs) sin 2ay«} e-40)«, (23) 
We notice how the presence of the smaller negative pressure complicates 
the mathematical expressions. On the other hand, all the terms are of the 
same type as in simpler cases; we have three terms involving the same 
exponential function, the third having an oscillating factor cos (2a, +e), 
where 
tane = (A2B,— AiB)/(AyAo+ Bi Be). (24) 
The humps and hollows on the curve for R will not coincide exactly with 
those obtained by graphing 
= +2) cos(Qaxe+e), with «= g/v, - 
but the agreement will be sufficiently close for present purposes. 
Accordingly, the maxima for R will be near speeds for which 
2ayce+e = nz; fe = 1b, Bs By sos 
The corresponding speeds and wave-lengths are given by 
pa Aue ; y= Estat, : (25) 
nit —e nT —€ 
In the previous case of symmetry, with the result in (19), the humps 
occur at wave-lengths 4a/n, that is when the wave-length is equal to or an 
odd sub-multiple of a certain length ; a similar statement in terms of velocity 
brings in the series J, 1/,/3,1/,/5, ete. In the present case we see from 
(24) that this arrangement is somewhat disturbed by the presence of the phase 
€, a quantity which may possibly be small compared with 7. A complete 
algebraical study might be made, but possibly a simpler way would be to 
start from a graph of the pressure curve and carry out the integrations 
involved in (8) by graphical methods. We can also obtain information by 
working out some numerical examples ; one may suffice at present, namely, 
&—180£+ 2419 
The pressure curve is of the form BC, shown in fig. 1, with 
h/H = 0541; 1 = 10°66. 
Further, with the previous notation, 
M = —a=5, db) = 4, be = ./34, 
101 
