PHYSICS AND MATHEMATICS TO GEOLOGY. 143 
there is no reason to believe that some of the limitations assigned here 
to the application of the mathematical theory will be accepted by all 
or even by a majority of the elasticians. In fact, the mathematical 
theory has actually been applied by several recent writers under cir- 
cumstances when most or all of the limitations proposed here are vio- 
lated. For instance, this is to a certain extent the case in Prof. Dar- 
win’s paper* “On the stresses caused in the interior of the earth by 
the weight of continents and mountains.” In the principal part of the 
paper he supposes 7=0.5, when, as we have seen, none of the objections 
apply; but in his § 10, in order “to know how far the results - - - 
may differ, if the elastic solid be compressible,” he supposes that while 
the rigidity constant is finite the bulk modulus is very small. In other 
words, he applies mathematical formula which assume 7 as nearly equal 
to —1. Such avalue has been here regarded as impossible. It should 
also be noticed that if 7) were equal —1 then E would vanish, and if 
7 be nearly —1 the value of EK must be very small. Thus the strains and 
displacements given by equations 2 to 4 would, in the case supposed 
by Prof. Darwin, be enormously greater than even those given in Table 
Iv. I do not observe, however, that either in the paper itself or in 
one supplementary? to it Prof. Darwin makes any explicit reference to 
the terms in the strain independent of the angular co-ordinates, from 
which the equations 1 to 4 are derived. I am thus unable to say 
whether his neglect of the limitations that these terms are here regarded 
as setting to the application of the mathematical theory is intentional 
or not. Again, in a recent papert ‘On Sir William Thomson’s esti- 
mate of the rigidity of the earth,” Mr. Love has also considered the prob- 
lem of the earth treated as an isotropic elastic sphere, more especially for 
the value of 0.25 of » In his equations 14 and 18 Mr. Love determines 
the values of two arbitrary constants which occur in the terms inde- 
pendent of the angular co-ordinates, and it is easily seen that the ex- 
pression he would thence obtain for these terms is identical with mine.§ 
After determining the second constant he however dismisses the sub- 
ject with the remark, “This - - - gives the mean radial displace- 
ment, a matter which need not detain us here.” So far as I can see, 
Mr. Love makes no reference to any principle such as C, nor to the 
possibility of the stress-strain relation ceasing to be linear. 
I ought also to explain that in my paper (a), directing my attention 
solely to the theories of rupture, I left out of sight any such limitation 
as (B) or (C), and treated the case of an earth in which 7=0 as one in 
which, according to the greatest-strain theory of rupture, the mathe- 
matical theory was applicable. I also failed to notice that the case 
n=0.5 was sanctioned by the greatest-strain theory as well as by the 
' stress-difference theory. 
* Phil. Trans., 1882, pp. 187-230. 
t Proceedings of the Royal Society, vol. XXxXvill (1885), pp. 322-328, 
{ Trans. Camb. Phil. Soc., vol. Xv, pp. 107-118. 
§ (a) Equation (17), p. 281, 
