492 TRANSACTIONS OF SECTION A. 



(lata given by Elsfcer and Geitel that if the earth contained throughout its volume 

 as much radium as a sample of clay examined by them the temperature gradient 

 at the surface would be about accounted for. I have recently examined a large 

 number of rocks, both igneous and sedimentary, and have been led to the con- 

 clusion that there is very much more radium iu all of them than would be 

 needed to maintain the earth's internal heat if the earth were constituted of rock 

 throughout. From this I conclude that the interior of the globe does not 

 contain radium, and that in all probability its composition is quite different in 

 other respects also from that of surface materials. 



My data for the quantity of radium in rock point to a thickness of at most 

 forty-five miles for the earth's rocky crust. Such a thickness of rock would 

 contain amply sufficient radium to maintain the earth's temperature gradient. 



Calculation on these premises, on the assumption that the thermal conductivity 

 of rock is not much affected by change of temperature, proves that the internal 

 temperature at the bottom of the crust (forty-five miles down) would be about 

 1 ,500° C. The inside nucleus, heated by the crust of radium-containing material, 

 must be at this uniform temperature throughout, just as a loaf of bread which 

 has been in an oven long enough takes up a steady temperature equal to that of 

 the walls of the oven, 



I have found in discussing the subject Avith scientific friends who are not 

 concerned with the detailed development of radio-activity that one objection is 

 specially felt. It is urged that a gramme of radium diffused through an enormous 

 volume of rock may not develop nearly so much heat as it would do if concentrated. 



In answer to this it may be replied that — (1) there is no reasonable doubt 

 that the heat development of radium is intimately associated with its peculiar 

 electrical behaviour. Indeed, according to Rutherford's data, it can be quanti- 

 tatively accounted for as the kinetic energy of the a particles emitted. The rate 

 of emission of a particles by pitchblende, as measured by ionisation, is exactly 

 what might be expected on the view that the radium atoms contained in the 

 mineral are as energetic as they would be if they were all collected together. 

 (2) Direct measurements made by Pegram on uranium and thorium have shown 

 that these feebly active elements give about the amount of heat which their 

 activity would lead one to expect. I think that these considerations leave no 

 reasonable ground for the objection above mentioned. 



Full details of the experiments and calculations here referred to will be found 

 in the Royal Society's ' Proceedings,' A, vol. Ixxvii, p. 472. 



TUESDAY, AUGUST 7. 



Department of Pure Mathematics. 

 The following Papers were read :— 



1. Some Notes on Finite Groups. By Harold Hilton. 



1. If Cayley's colour-groups are to be of practical value they must be suf- 

 ficiently simple to appeal readily to the eye. This can often be secured by drawing 

 the diagrams on a surface of suitable class. In particular, diagrams for finite 

 groups of ' genus one ' can be drawn on an anchor- ring ; and these diagrams give 

 a simple method of determining the order and generating relations of all such 

 groups. 



2. A group of order p"^ whose group of self-conjugate elements is of order ^* con- 

 tains an Abelian sub-group of order />^'+^, if 2a>3(2c + /3-l). This theorem is 

 an extension of one due to Dr. G. A. Miller, and can be proved by the same 

 method. 



u. \i a,h are elements of the Sylow sub-groups of order ja", y* of a group G 

 of order p'q^r'' . . ., we can readily prove that a and l> are permntable if the 

 corresponding elements of the group of po-gredient isomorphisrns are permutabje, 



