4n0 



Tlie possibility of the water of crystallization acting tlirongh the 

 increase of the moment of inertia whicli it produces was previously 

 expressed by Kamerltngh Onnes and Perrier. They worked on the 

 view that the deviations were due to the PLANCK-vibrators comino- 



o 



to rest and that their frequency would be smaller for the molecule 

 of crystallized gadolinium-sulphate than for the unloaded molecule. 

 By comparing anhydrous and crystallized manganese sulphate an 

 estimate may be formed of the distance between the centra of the 

 water-molecules and of the sulphate molecule. The moments of 

 inertia of Mn SO, and of Mn SO, . 4 H,0 are found equal to 

 8.7 XIO--*», and 109.7X10-^'. On the supposition that two of 

 the water-molecules are placed along a line at right angles to the 

 axis of rotation and the other two on the axis, the difference of the 

 two above numbers gives for the distance in question the value 

 4,4 X 10"^. This distance is of the order of the molecular dimen- 

 sions, but smaller than the adopted value for the radius of a 

 hydrogen molecule (IX'-IO-^): our result w^ould thus seem to 

 indicate that the water-molecules partly penetrate into the molecule 

 to which they are attached. 



§ 7. At the suggestion of Dr. Keesom — to whom I am indebted 

 for several useful hints — I have also compared the observations 

 with a different formula which takes into account the possibility of 

 more than one frequency of rotation, whereas Einstein and Stern 

 assumed that at a definite temperature all the molecules rotate with 

 the same velocity. On Dr. Keesom's theory tlie molecular rotational 

 motions in the body under consideration are analysed into a number 

 of stationary waves, which may be supposed to be governed by 

 similar partial differential equations as, say, sound waves in a gas 

 with corresponding conditions at the boundary (crests may take the 

 place of nodes). Along these lines he was led to the hypothesis, that 

 equation (4) of Suppl. N". 30^z (May 1913) with the additional assump- 

 tion v,n ^ IJi (as in equations (5) and (7) 1. c. ) might also hold for 

 the rotational motions. I have found that the correspondence between 

 observation and calculation by the new formula is not appreciablj^ 

 better than by the method followed above and I have therefore in 

 this paper confined myself to the calculation according to the Einstein- 

 Stern view i. e. assuming one value of v at each temperature. 



§ 8. Swinmari/. The existence of a zero-point energy has recently 

 been made probable by various investigations. 



1. The change of the specific heat of hydrogen at low temperatures 



