484 Lord Rayleigh on the Open Organ- Pipe 
Confining ourselves for the present to the case whereithe 
total width of the strip is small compared with the wave- 
length, we. have to integrate the second series in (6) with 
respect to 2, for which purpose we have the formula 
kt 1 : 
{2 los was — ay | log e— md § : (8) 
In this way we obtain V, the potential at the edge of the 
strip of width z. Afterwards we have to integrate again 
with respect to y and zw. The integration with regard to y 
introduces simply the factor y, representing the (infinite) 
length of the strip. The integration with respect to # is 
again to be taken between the limits 0 and «. Thus 
SPV de dy = —yo?{y—$+log(sike)}, . . (9) 
terms in 2 being omitted. This is the equivalent of (3) in 
the case of an infinite strip of length y and width «@. 
Accordingly by (2) 
J) 8pdS=— —— g, Yt {y—3 + log (Shur) + diz}. (10) 
The reaction of the air upon the plate may be divided into 
two parts, of which one is proportional to the velocity of the 
plate and the other to the acceleration. If & denote the dis- 
placement of the plate at time t, so that d&/dt=dd/dn, we 
have 
Ge de. ie 
dP =tha 7 =tka7 : 
and therefore in the equation of motion of the plate, the 
reaction of the air is represented by a dissipative force 
d& ac 
LY Te ke... . > 
retarding the motion, and by an accession to the inertia 
equal to ) 
o : ha 
Ty... »(g-y- log"), = 
the first factor in each case denoting the area of the plate. 
The mass represented by (12) is that of a volume of air 
having a base equal to the area of the plate and a thickness 
tr a ka 
“(-y- log 5) ae 
When « is given (13) increases without ican ~when 
(=2/k) is made infinite, as we found before. But if we 
(13) 
