120 Mr Glazebrooh, A comparison of Maxwell's [Feb. 25, 



(iii) 



E(n) 



-2E (n -1) + 2E(n - 2) 



-SE(n-S) + SE(n - 4) - SE (n - 5) 



+ 4>E(n-6)- 



+ (-) n sE(0) = 0, 



where E (0) denotes \ or according as s is even or uneven. 



In the cr-formula the groups of terms have alternately positive 

 and negative signs, in the ^-formula all the terms are positive, in 

 the jE'-formula the signs of the terms are alternately positive and 

 negative. 



(2) On primitive roots of prime numbers and their residues. 

 By A. R. Forsyth, B.A. 



(3) A comparison of Maxwell's equations of the electro- 

 magnetic field with those of Helmholtz and Lorentz. By R. T. 

 Glazebrook, M.A., F.R.S. 



The equations of the Electromagnetic Field have been de- 

 veloped by Maxwell (Electricity and Magnetism, II.) on the one 

 hand and by Helmholtz (Borchardt's Journal, Bd. lxxii. Ueber 

 die BewegiXngsgleichungen der Electricitdt) and H. A. Lorentz 

 (Schlomilch Zeitschrift, xxii.) on the other, starting from somewhat 

 different standpoints. The object of the present communication 

 is to give a comparative account of the two theories in the 

 endeavour to discover the fundamental differences which lead 

 to the different results actually arrived at. 



According to Maxwell, when electromotive force acts on a 

 medium, electric displacement is produced through it, the medium 

 being polarized, and if P be the total electromotive force in the 

 direction of the axis of x, the medium being isotropic, f the 

 electric displacement and K the inductive capacity, we have 



/= ZP/47T. 



Suppose we take any section of the medium normal to the 

 axis of x ; then we know we can distribute electricity over that 

 section so as to produce at any point of it the actual resultant 

 electromotive force normal to the section, f the electric displace- 

 ment, will be the surface density of the electricity so distributed 

 and to quote Maxwell {Electricity and Magnetism, Vol. I. § 62) 

 " Whatever electricity be, and whatever we may understand by a 



