419 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
Further, when /” (wv) is zero, we have 
FT (w) = Ageu (2cu— 3a cos a)/(w?—2cus cos a+ cu?)3, (79) 
Using the formula (17) we obtain the particular group of terms from the 
integral (72) as 
pan, ue Qa \ 3 
= ~ Ber (w? — 2eum cos a+ 2u?)? iF ih CEN) ce) 
in which the special value of w must be substituted. 
Evaluating this expression we obtain 
pee of {3 cos «+(9 cos? a—8)"}4 
2* pc w* (9 cos?a—8)* {3 cos? a—2+4 cos « (9 cos? oy 
1a gu/ 2 {8 cos a+(9 cos? a—8)*}? —in} (81) 
16 {3 cos?a—2+4 cos a (9 cos? «—8)'}4 
This represents a system of transverse waves travelling with the 
originating impulse ; the amplitude for a given azimuth « diminishes as o7?, 
On the central line, where « is zero, this reduces to 
93% 
B= a (G-1), 62) 
T pew G 
corresponding to simple line waves of length suitable to velocity ¢ on deep 
water, but with the amplitude factor o~?. 
Following the crest of a transverse wave we have 
gean/2 3 cos a +(9 cos? « — 8)*}? —hor = (2n+1)z, (83) 
16¢? {3 cos? a—2+4cos «(9 cos?a—8)*}4 iow 
where 7 is a positive integer. The crests cut the axis in points given by 
w= (2n+4)r/9, (84) 
and cut the radial boundaries given by « = + cos 2,/2/3, in the points 
w = 2c? (2n4+4) r/g,/3. (85) 
Consider the variation of amplitude following a crest; we substitute for a 
from (83) in (82) and obtain 
const. {3 cos a+(9 cos? a—8)*}# 
(2n+ 5) (9 cos? «—8)*{3 cos? a —2+ cos « (9 cos? a—8)*}? 
(86) 
This becomes infinite at the outer boundary, when « is approximately 
19° 28’; this is due to the failure of the method of approximation and we 
shall consider it later. For the present the following table of values and 
curve show that the approximation holds up to angles very near the limit. 
22 
