160 
Proceedings of the Royal Society of Edinburgh. [Sess. 
function of a, half the arcual distance from epicentre to point of emergence, 
or, symbolically, 
(9) 
In this expression p is given as a function of a ; but the relationship 
may be supposed to be inverted so that a is given as a function of p, or, 
symbolically, a=f(p). The form of this function cannot be expressed 
analytically, but by suitable treatment of the observations it may be 
represented by a series of tabulated numbers to any degree of closeness 
that may be found to be necessary. 
The integral equation (5) then becomes 
(10) 
from which it is required to determine the quantity v as a function of r. 
Putting v/v = tj, a new variable, so that 
— = d 0°g r) = ~ (log r)cl v , 
r Crj 
we get the integral equation in the form 
.i/v a 
(* I- (log r) *7 
f(p) =p I y=i= 
Jv 2 ~P 2 ’ 
where V is the value of v at the surface r= 1. 
As shown by Bateman, this may be identified with an integral equation 
solved by Abel. The solution is 
»(I ogr) — *4 r^P 
Jri 
dr, 
Vo Jp 2 -v 2 ’ 
or 
log r = C - - 
2 f 1/v J 
f(p)dp 
x/p 2 - v 2 ' 
At the upper limit p= 1/V and log r = 0. Hence C vanishes and 
log T 
*_ _ 2 f 1! 
TTJ-n 
1 ApYp 
Jp 2 -7] 2 
( 11 ) 
By means of equation (10) or its equivalent a solution may be obtained 
by assuming some convenient expression for v 2 which will make the 
integral immediately integrable. This was the method adopted in my 
earlier paper, and it was also essentially the method adopted by Wiechert 
and Zoppritz. Even Bateman, after developing the solution as given in 
