286 



If the susceptibility does not depend on the field and its topo- 

 graphy remains the same, the enrves are straight lines. The small 

 deviations from the straight line at 20.°3 K are probably chiefly due 

 to errors in the topography of the field, seeing that according to 

 earlier more accurate determinations we had to await within the 

 limits of the experiments a susceptibility independent of the field and 

 therefore in this graph a straight line. By means of the deviations 

 from the straight line at 20.°3 K the curves for the other tempera- 

 tures have been provisionally corrected. It will be seen that for a 

 given field these curves are the more strongly curved the lower the 

 temperature to which one descends, in accordance with Langevin's 

 theory. Within the limits of accuracy to be expected in connection 

 with the neglect of the various corrections referred to above the 

 tangents of the angles of elevation of the tangents to the curves at 

 the origin appear to be inversely proportional to the temperature as 

 required by Curie's law and the deviations of the curves from the 

 tangents as expressed by the ratio between the ordinates of both 

 for a given abscissa are strikingly similar to the deviations of 

 Langevin's curve for the magnetisation as a function of the field 

 expressed in the same manner. The nature of paramagnetic magne- 

 tisation is very clearly revealed in these measurements at helium- 

 temperatures. 



Mathematics. — "(M some integral equations." By W. Kapteyn. 



1. In a memoir "Recherches sur les fonctions cylindriques" (Mém. 

 Soc. Roy. Sc. Liege 3'^'"^ Série t. VI 1905) we gave the solution 

 of the integral equation 



ƒ(..)==ƒ/(,?) 7o(.t—/i)<>' • (1) 







in this form 



,(..)= I' +r/v)^-|^/- .... (2) 



d.v J .V — p 







where the functions fk represent Bessel's functions of order k. 

 This solution rests upon the relation 



j^'<«-^^<''^=— —(^i. = 0.1.2.. j •••(«' 







from which the following theorem may be deduced. 



