shaft turned by the handle indicates that it has turned through an angle proportional to 2.405, which is the first value of a which satisfies Jo(x)=0.

It is advisable to have a pointer and a scale to indicate exactly the displacement of the table, so as to test the accuracy with which the cam performs its duties. Of course, when the graduated-circle indication is 2-405, the displacement of the table is to be 2.405 J1(2405) or -1.249 inches. The area recorded on the planimeter in square inches must now be multiplied by 2(2·405)2/a* [J1(2·405)]2, and the answer is A1.

To find A, change the gearing so that when the whole roller-curve passes under the tracing-point of the planimeter, the graduated circle indicates 5.5201, and check the error of the cam by noting that the displacement indication ought now to be 5.5201 J1(5·5201) or 1.878 inches. The area recorded by the planimeter in square inches must now be multiplied by 2(5.5201)2/a [J1(5.5201)]. If variable frictional gearing is used, it is important that the roller should be placed on roller bearings of small resistance.

To develop an arbitrary function in Bessels of any other order, or in Fourier's Series, or in zonal harmonics, or in series of functions of any other normal forms, we have only to replace the cam by one of another shape; so that this one simple machine is suited to quite general analytical use.

XI. On the Effect of Sphericity in Calculating the Position of a Level of no Strain within a Solid Earth, and on the Contraction Theory of Mountains. By Rev. O. FISHER, M.A., F.G.S.*

I

HAVE been permitted to reply to Professor Blake's criticism upon my investigations concerning the relative structure of the continental and suboceanic crust†, and I now hope to do the same to his objections to the calculation of the depth of the "level of no strain" -a subject which he admits to be important.

Mr. Blake says that he is surprised that the superficial position of the level of no strain at no more than four miles from the surface "should not be regarded as a reductio ad absurdum that the method or premisses which lead to it must be wrong, both a priori, that any critical change in condition

*Communicated by the Author.

† Phil. Mag. vol. xxxvii. p. 375, April 1894.
'Annals of British Geology,' 1892, p. iv.

could be demonstrable at so insignificant a depth, and a posteriori, considering the magnitude of the actually observed features of the earth's surface."

As regards the a priori objection, that it is improbable that any critical change in condition should be demonstrable at the small depth of from two to four miles, it may be replied that no claim is made that such a change in condition exists at that depth in the actual earth. But what has been demonstrated is, that, if the earth had cooled as a solid globe, a level of no strain would be found in that position. In that case, however, the surface-features would not have resembled in size and arrangement those which we see. Consequently we do regard the improbability of a critical change at so small a depth to be a reductio ad absurdum, and conclude that one of the premisses, viz. that of solidity, is wrong. The a posteriori objection involves a petitio principii; for it assumes that the observed features are due to contraction through cooling, which is the very question that is being brought to the proof.

But it is in the method of investigation rather than in the premisses that Mr. Blake appears to think a mistake has been made; for he contends that the use of the "linear"* equations for the conduction of heat in the calculation of the position of the level of no strain, as was done by Mr. Davison and Professor Darwin †, and also by me, is inconsistent with the introduction of the radius of the earth already assumed infinite. The objection is primâ facie plausible, and it had occurred to me; but seeing that the changes of temperature involved occurred only near the surface, I did not think it necessary to take sphericity into account as regarded the temperature gradient; and what follows will show that I was justified. Using for convenience of reference the same symbols as in my book :

r = the radius of the earth considered spherical, 20,902,404 feet, 3958.78 miles ;

t = the time since the globe solidified ;

V the temperature of solidification;

=

the distance of a spherical shell from the surface;

z = distance of the same from the centre ;

v = the temperature of that shell when sphericity is neglected as was done in the work referred to;

* Mr. Blake thus refers to the equation for the conduction of heat in one dimension.

Phil. Trans. Roy. Soc. vol. clxxviii., 1887.

Physics of the Earth's Crust,' chap. viii., “On the Cooling of a Solid Earth," 2nd ed., 1889, p. 94.

u = the like when sphericity is taken account of;

e

the coefficient of linear contraction of those portions of
the globe which have sensibly cooled ;

x = the conductivity measured in terms of the capacity of
rock for heat;

then the differential equation for the diffusion of heat in the sphere will be

Professor R. S. Woodward, U.S.A., gives the solution of this, which is suitable to the case of a sphere initially at a uniform temperature throughout, and cooling into a medium such that its surface is maintained at a constant temperature considered to be zero. The solution is *

This solution meets the objections raised by Professor Blake to Lord Kelvin's solution, in which the radius was assumed infinite.

Prof. Woodward, with great ingenuity, transforms the above expression into one or other of two rapidly converging series, in either of which he says the first term is sufficient in the case of the earth, if the time since the commencement of the cooling is less than 100,000,000,000 years. The second of these series (no. 20) written in our symbols is

The first term of this series being sufficient, if we differentiate it with respect to t we get

*Annals of Mathematics,' vol. iii. June 1887.

The convergency of this series is evidently due to the rapidity with

S

0

e-2 du approximates to the limiting value

Tas the upper limit increases. When that is no greater than 2:17, the first six places of decimals are the same as for the limit, and when it is 4 the first ten.

This differs from the expression for the corresponding time rate (dv/dt) in the case when sphericity is neglected by being multiplied by the factor r/(r-x). Hence the fall of temperature at a given depth goes on slightly more rapidly in the case of the sphere, as might be expected. Again, differentiating with respect to a, we have

Putting a=0, we get for the temperature gradient at the

When sphericity is neglected, or r infinite, we have

2 V NTT B'

√ Art= =a of Lord Kelvin's problem of secular cooling, and as in my 'Physics &c.' Hence the time which elapses before a given surface-temperature gradient is acquired is somewhat shorter when sphericity is taken account of, as might also have been expected.

Lord Kelvin assumed the high value of 7000° F. for V, the temperature of solidification, probably to allow for its possible increment in the lower shells owing to the pressure. With this value a=402,832 feet. But, when sphericity is taken account of, 4kt becomes 396,073 feet, which makes it 1 mile and 1497 feet less.

The equation which gives x, the depth of the level of no strain, is *

If we substitute the value just found for du/dt, and take 4kt for the unit of length, this may be reduced to the

*Physics, &c.,' p. 95.

S

-x2

following equation :

(r−x)(2(r−x)x− 3)+3€¤2

dx=0.

If we make a zero, the first side of this becomes negative. If we give such a value as will make the first term vanish, it becomes positive. This value is (r-√72—6), or to our present unit is 0.028. Hence the value of x which gives the level of no strain lies between 0 and 0.028, so that x2 is small, and 3 and higher powers may be neglected. We may therefore put

The equation then becomes

(r−x) (2 (r−x)x−3)+3(1+x2) (√1⁄2′′ −x)=0 ;

whence, neglecting x2/r2,

2

This differs from the expression when sphericity is not considered, in the small terms. With the values 1/51° F. per foot for the gradient at the surface, and 7000° F. for the temperature of solidification, the depth of the level of no strain was found to be 11,252 feet *. But the depth when sphericity is taken account of in the cooling will be 11,071 feet; so that the level of no strain is brought nearer to the surface by 181 feet by this consideration. The resulting difference, however, comes out so small as amply to justify sphericity being neglected, as was done by Professor Darwin and myself.

It is obvious that the corrugations formed by compression

* 'Physics &c.,' 2nd edit. p. 98.