58 MODERN SEISMOLOGY 
If the law of variation of v with v is known we could evaluate 
the integrals. We do not, however, know this law, and the 
problem before us is whether, from the graphical representation 
of T as a function of 4 or @ from observations, we may deter- 
mine v as a function of 7. 
The analytical solution is expressed by 
a y el so 
Hel = hy | Co Oe 
(cf. Bateman, “ Phil. Mag.,”’ 1910), and 
I 
2 8 6 [” 2! 
Te ee a se Ge) eine 
so that if @ and T can be expressed as functions of ¢ or a 
we should get 7 as a function of » and hence the velocity at 
any depth. Now the observations give T as a function of 
A, so that theoretically the problem is solved. But asa 
matter of fact time curves are still very inaccurate and do not 
justify a very minute analysis at present. 
One must proceed by a comparatively rough graphical pro- 
cess, and the obvious suggestion would be to take successive 
ranges within which @ does not vary much with c. 
Wiechert, who first attacked the problem, divided the Earth 
into finite layers within each of which the radius of curvature 
of the path might be taken as constant, and on this basis 
Wiechert, Zoppritz, and Geiger (l.c.) analysed the time curves 
for Pand S. The results of the investigation which are set out 
in the table, page 61, show that from 4=0 to 4 = 5000 km., 4,, 
increases from 0 to about 1500 km., while V, and V, continually 
increase as #,, increases. As J increases to 6000 km. &,, 
increases very little. Beyond this 4,, again increases until for 
4=13,000 km. 4,, attains a value rather over 3000 km. But 
from 4,,= 1500 to 3000 km. both V, and V, remain constant. 
It is specially interesting that Poisson’s ratio o remains practi- 
cally constant. 
The variation of velocity with depth may not, however, be 
continuous, but we may have surfaces at which the velocity 
undergoes a sudden change. Such a surface of discontinuity 
