403 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
§ 4. Features of the Integrals Involved. 
The integrals we have to consider in such problems are of the type 
= | $ (u) cos { f(u)} du. (11) 
All such integrals we can treat in the same manner, adopting the method 
employed by Lord Kelvin for the particular case referred to above (§ 3). 
This method consists in supposing that f(w) is large, so that the cosine factor 
is a rapidly varying quantity compared with the first factor; thus, much as in 
the Fresnel discussion of the diffraction of light waves, the prominent part of 
the graph of the integral is contained within a small range of w for which 
7(u) is stationary in value, so that the elements are then cumulative. In 
other words, we select from (11) the group or groups of terms ranging round 
values Uv of uw which make 
f (%) = 0. (12) 
In such a group of terms we may put 
Su) = FU) +3 (w= wo)? P" (Uo). 
Then if we write o? for }(w—w) jf’ (wo). the contribution of the group to the 
value of (11) is given by 
yo = {ar} | 6 (uw) cos { f (uo) + 02} do, (13) 
where the limits of the integral may be in general extended, as in diffraction 
theory, to +00, provided w does not coincide with either limit of the integral 
(11), and also provided that /’’(w) is not zero. 
Thus we have, from (13), 
w= { wale (uo) [eos {f (vo)} sin {F (wo) }] 
=f 57 TGR mt $ (uw) cos { f (uo) +477}. (14) 
This is the sum of the contributions of the constituents of each group 
around a central value w given by (12), provided the value % comes within 
the range of values of uw in the integral (11). 
If 7’’(w) is negative, the Sere a result may be written 
Wo = | rap f # le) c05 {/ e0) 4}. > 
We write down for reference the similar pair of results for a group of terms 
from the integral 
— | (w) sin {f(u)} du. (16) 
