June 28, 1906] 



NATURE 



199 



tralc into the interior, and electric curents circulating in 

 the sphere in parallel planes will cause a magnetic force 

 op|)i)sL-<l to that due to the inoving charges. If the con- 

 ductivity be perfect, these currents will persist, and the 

 intcriiir of the sphere will be permanently free from either 

 clcitric or magnetic force; but, with finite conductivity, 

 the lurrents will die away, and the magnetic force will 

 tinally attain a definite value inside the sphere, although 

 the ileitric force vanishes. In each of these cases a solu- 

 tion is easily found ; but while the currents are dying away 

 the magnetic energy gradually changes, and the calculation 

 of the energy at any given time might be difficult. 1 

 thercfure confine myself to the two limiting cases. The 

 case of the final stage when there is finite conductivity 

 I h.KJ solved when Dr. lieaviside suggested to me that 

 1 should include the case of infinite conductivity in my 

 investigation. The present communication is the outcome 

 of that suggestion. 



The sphere is of radius a, and carries a charge Q at a 

 speed u, which is very small compared with v, the velocity 

 of light ; the strength of the field is F, and the specific 

 inckiitive capacity is c. I employ Dr. Heaviside's units 

 in order that my results may be comparable with his. 

 The origin is at the centre of the sphere, and the axes 

 of A and y are respectively parallel to the direction of 

 motion and to the direction of the uniform electric field. 



When ulv is very small, the electric force E, due to the 

 mnviiig sphere, is the same as if the sphere were at rest, 

 and is therefore derivable from a potential function. 

 .Since there is no electric force inside the sphere, the 

 induced distribution on the sphere produces the potential 

 Fy at internal points, and hence produces the potential 

 Fa^yfr' at external points. Thus at internal points the 

 components of E are 



Ei=o, E2=-F, £3 = 0, 

 while at external points they are 

 Q.r 



E,= 



4-jrCf 

 E,= ^~ , + Fa- 



K?-:^)' 



.^1'= 



Finite Conductivity, Final Stage. — In this case the mag- 

 netic force is entirely due to the motion of the charge on 

 the sphere, and we have 



H]=o, H2= - ;(fE.„ H2=ii(En, 

 and thus H^ = «^c^(E,^-l- E,'). 

 If, now, we write 



X ^ ;- cos 0, y = r cos 1^ sin 0, z=-r sin <(> sin 9, 

 we find that for internal points H^ = u-c'F^, and that for 

 external points 



H^ = «-c- ., .) i + -^ -' cos (() sin 9(3 sin- fl - i) 



Ll6Tr-£"-;"* 2Tr("; 



-I- - - ,'. I 1-6 cos- (^ sin- 9 + 9 cos- <(> sin-i 9 j 



The magnetic energy is \ii.W' per unit volume, and thus 

 the total magnetic energy is 



■\ = \^tfic^¥"-.%-^ci' + \y. 



/./"/— 



dr (19 d(p. 



where r ranges from a to infinity, ^ from o to 27r, and 

 9 from o to ir. On effecting the integration, we find 



T >= \u^ [/iQ-/6Tra + 1 65r^t-F-aY5] = \mir, 

 where m is the magnetic inertia. 



For motion parallel to F instead of right angles to it. 

 Dr. Heaviside finds (Nature, April 19) 



T = \i<- {^iq-l6ira + 87r^-F-a3/5]. 



When the quantities are measured in ordinary units the 

 results become 



T = \h- [2;aQ-/3a + 4M'-'F^«7S] (Searle ) 



and 



T = W- [2|uQ=/3(j + 2iic-F~a^/sl (Heaviside) 



NO. I913, VOL. 74] 



where, if we use C.G.S. electromagnetic units, we have 

 /i=i and £ = (3X10'°)-°. 



Infinite Conductivity. — In this case a system of currents 

 flows round the sphere, on its surface, in planes normal 

 to the axis of z, the distribution being such as to give 

 rise to a magnetic potential —ucFz at internal points. 

 The magnetic force due to these conduction currents then 

 neutralises, at internal points, the magnetic force —ucF 

 due to the moving charges. The external magnetic force 

 due to the conduction currents will satisfy Laplace's 

 equation for very slow speeds, and must, therefore, be 

 expressible in zonal harmonics, while, for all speeds, the 

 magnetic force normal to the sphere must be continuous. 

 If s/r = cos <li, the normal force at points just inside the 

 sphere is ucF cos .f'. The conditions are satisfied by the 

 external potential 



n = iicFa^ cos <ii/2i-- = lie Fa^z/2r', 

 for this is a zonal harmonic, and gives rise to the normal 

 force ucF cos it at the surface. 

 Thus, at external points, 



Hi=-dajdx, H^=~dn/dy-ucE3, n^= - dajdz + iicF^, 

 where E,, E., Ej have the values already given. We thus 

 find 



TT 'i.ucFd'xz 



Hi = — ^^^. 



H — - "''Q^ _ y'cFahiz 



H3 = 'i^+«.F«3p-'''t3-_.Al. 

 4irrr' L 2r^ 2r^J 



It will be found that «H, -l-yH,-+-zH3 = o for all values 

 of r, and thus the magnetic force is tangential to the 

 sphere, a condition pointed out to me by Dr. Heaviside. 

 Hence we find 

 H2= „2,2 rOisin^S + 3QF«' eos * sin 9(2 sin'^9 - I ) 



+ — '^''{36sin^9cos-<j>-9sin-9(4COs-if>-t-sin-» + 9^- • 



Remembering that H=o at points inside the sphere, we 

 find, on integration through the external space, that the 

 magnetic energy is 



T = 4«- [MQ767ra + e^/xc-FVIsl 

 When the quantities are expressed in ordinary units the 

 result becomes 



T = hr [2mQ73" + 3Mf''F''a'/ 10]. 



If an electron be a conducting sphere of radius 10-" cni. 

 with a charge of 10-"° electromagnetic units, an electric 

 force of a billion volts, or io-° C.G.S. units, per centi- 

 metre, would not change its inagnetic inertia by so much 

 as one part in ten billions, and the results are of no con- 

 sequence in experiments on the electrostatic deflection of 

 kathode rays ; but it is possible that there are other cases 

 where it would be necessary to take the change of mag- 

 netic inertia into account. 



When « becomes comparable with v, the analysis be- 

 comes more complicated, but does not present any diificiilty, 

 at least in the final stage, with finite conductivity, provided 

 a Heaviside ellipsoid be substituted for a sphere. 



In conclusion, I desire to acknowledge the help I have 

 received from Dr. Heaviside's suggestions, and to thank 

 Mr. Norman R. Campbell for verifying the formulae. 



G. F. C. Searle. 



The Date of Easter. 



That the formula of Gauss for finding the date of 

 Easter fails in certain cases, of which the year 1954 is 

 one, was pointed out by Gauss in his original paper in the 

 Monatliche Correspondenz (vol. ii., p. 129), where he shows 

 that there are the following two exceptions to the formula 

 in the Gregorian calendar : — 



(0 When the formula gives April 26, Easter falls ore 

 April 19. 



(2) When the formula gives d = 2S, e = 6, while iiM-l-ii 

 divided by 30 gives a remainder smaller than ig, then 



