164 
Proceedings of the Poyal Society of Edinburgh. [Sess, 
values of f(p). One of the quantities p 2 is then chosen as the limit and 
subtracted from all above it, and the square root of all these differences 
taken and tabulated. Each f(p) is then divided by the corresponding 
Table II. — Secondary Wave. 
pjl. 
8a 
f(p) = a/2. 
P « T . 
8a 
f(p) = a/2. 
0T 
F da 
f(p) = a/2 . 
28 
0 
20 
11-4 
12 
35-5 
27*5 
1-25 
19-5 
12-0 
1T5 
39-8 
27 
2-35 
19 
12-6 
11 
430 
26-5 
3*3 
18-5 
13 25 
10-5 
46-0 
26 
4-2 
18 
13-95 
10 
48-5 
25-5 
5 
17-5 
14-7 
9-5 
51-0 
25 
565 
17 
153 
9 
53-5 
24-5 : 
6*3 
165 
15-85 
8-5 
56 0 
24 
6*85 
16 
16-35 
8 
58-5 
23-5 
7-4 
15-5 
17-1 
7-5 
61-0 
23 
8-0 
15 
182 
7 
63 5 
22-5 
8-6 
14-5 
19T 
6-5 
66-0 
22 
9T 
14 
20-0 
6 
68-5 
21-5 
96 
13-5 
21-8 
5-5 
61-0 
21 
102 
13 
23-1 
5 
63-5 
20-5 
10*75 
12-5 
26-5 
value \/(P 2 — k 2 )> an d this when multiplied by' dp is one of the elements 
of the integral to be finally summed. 
The process is shown in Table III, in which 14 is the lower limit 
value of p. 
(3) The Evaluation of the Integral and of all the other Quantities 
involved . — In this table the quantity dp has the value 0*5 for all the 
intervals except the last five. For these it becomes 01, a diminution 
which is necessary, since the values of f(p)/V(p 2 ~~ V 2 ) r i se rapidly towards 
infinity* as 's/ip 2 — diminishes towards zero. The curve giving the 
relation between f(p)/V(p 2 — f) an d p is shown in fig. 4 (p. 162), fig. 3 
being the graphical representation of \/(p 2 — 14 2 ) in terms of p. What 
is sought for is the area of the region between the curve of fig. 4, the 
vertical axis and the horizontal asymptote. The greater part of this area 
can be readily reckoned by the process of mechanical quadrature. Leaving 
out of consideration meanwhile the part from p = 14 to p = 14T which is 
bounded by the asymptote and the infinite branch of the curve, we first 
calculate the area of the portion from p= 14 1 to p= 145, using the formula 
area = 1 4 K + + 64 (“i + u z) + Uu 2 h 
45 
where u 0 , u v u 2 , u z , u 4 are the five values of the ordinates bounding the 
four elemental strips and h is the width of the strip, in this case 0T. 
