TRANSACTIONS OF SECTION A. 577 



armature, and on to a circular scale. Varying currents -were sent round the field- 

 magnet coils, producing different values of induction in the armature. 



At a low induction the hysteresis is small and increases but slowly, the range 

 corresponding with the first part of the B/H curve, and the shape of the curve 

 being similar to that given by an alternating field. With increasing induction the 

 value of the hysteresis rises more rapidly, but does not continue to increase at a 

 aniform rate. When B = 12,000 the rate of increase of the hysteresis diminishes, 

 though the curve still rises rapidly. When B = 17,000 the curve begins to bend 

 over, reaching a maximum at 18,000 or 18,500, and rapidly falls when the induc- 

 tion is further increased, until when B = 21,000 the hysteresis is about one-third of 

 its maximum value. Higher inductions could not be satisfactorily reached with 

 the apparatus, but there was no indication of any diminution in the rate of decrease 

 of the hysteresis. 



Both soft charcoal iron and hard high carbon steel were tested, and found to 

 give the same results, the value in the steel reaching a maximum at an induction 

 of 16,000. 



These experiments show that there is no simple relation between hysteresis and 

 induction. The three stages of magnetic displacement each have a sharply defined 

 position on the hysteresis curve, the critical value occurring just on the knee of the 

 induction curve. The agreement of the results with the theoretical deduction 

 constitutes a strong verification of the molecular theory of magnetism. 



The above results hold good for all speeds up to seventy revolutions per second, 

 which was the highest speed obtained. 



13. On the Vibrations of a Loaded Spiral Spring. 

 By L. R. WiLBERFORCE, M.A. 



The author pointed out that by comparing the two periods of vibration of a 

 body attached to a spiral spring of small angle the ratio of the torsional and 

 flexiiral rigidities of the wire or strip forming the spring could be found, and 

 hence, if the wire were homogeneous, isotropic, and of circular section, Poisson's 

 ratio for its material could be determined. 



Some experiments were shown illustrating the normal modes of vibration of 

 such a system, and the transference of energy from up-and-down to twisting vibra- 

 tions, and back again, which can be effected when the periods are nearly equal. 



TUESJDA Y, AUGUST U. 

 The Section was divided into three Departments, 

 The following Papers and Report were read : — 



Depaetmeiti I. 



1. On Fuchsian Functions. By Professor Mittag-Leffler. 



Professor Mittag-Lefller, after referring to a recent investigation by one of his 

 pupils, M. G. Oassel, spoke of the advantages and inconveniences connected with 

 the expression of automorphic functions by means of the Fuchsian or Kleinian 

 6 series. He showed how the theory of the 6 functions may be brought into 

 connection with that of the Abelian functions, and that an expression given by 

 M. Schottky for a certain ckss of automorphic functions is, in fact, applicable to 

 all these functions. He then referred to his own researches on the invariants of 

 linear differential equations, and pointed out that Giinther, in his paper in ' Crelle's 

 Journal,' has not fully realised the true nature of these researches. 



1894. P P 



