580 T. H. Havelock. 
with the asymptotic expansion 
eee (5+ 25-(z yr 43 (v-3)+ 151 (v-2) b] 39 
Rw~ 3+ in (Op un pa iGsal oe D7 . (32) 
For small values of p, (81) is not satisfactory as the quantity within large 
brackets is the small difference between large numbers; it is better then to 
use an ascending series in p which can be found by substituting expansions of 
Yo 1, Yo and Y;. ‘The first few terms are 
Bee ee aon | 2 
R= 72 gpl | (Faye a760” + log ( 2 y 
Tas 
+576? 930400” + rf (2) 
where y is Euler’s constant, 0°57722 
From these expressions the values in Table I have been calculated. 
Table I.—Resistance of Model A. 
P- e| V (gL). 10°R/gpb2U. 
p- | e/ (gL). | 10°R/gpb7U. 
19 “64 0° 7°8 0 °358 56 
18 07 @)e 7 086 0-376 | 131 
16°5 0:2 6 0-408 406 
14°92 0:2 5 0-447 897 
14, O° 3° 0 5038 1502 
13 °36L O° 2 0-707 2392 
11°78 O° 1 1:0 2050 
10°22 Ov O° 1-058 1930 
9°24 Otsy 0° 1414 1434 
8 64 O° Os 2°0 904 
A certain portion of the range will be studied in detail later; the Table 
gives a general view of the variation of the resistance with the velocity. At 
low velocities the resistance is small and oscillates in value; then at a speed 
of about 0-4,/(7L) it begins to rise rapidly and reaches a maximum at about 
/(4gL), after which the resistance decreases continually and converges to 
zero for infinitely large velocities. Doubtless the conditions under which the 
expressions were obtained would be violated at very high velocities, but it is 
of interest to trace the variation in value over the whole range. Absolute 
values could be obtained from Table I for a plank of given dimensions and of 
the specified form ; these would be comparable with experimental results for 
a plank of finite depth if the velocity were such that the effect of the surface 
waves could be neglected at the depth of the lower edge of the plank. 
8. For Model B we take 6=1. The formule are now 
208 
