328 Mr. 0. Heaviside on the Electromagnetic Effects due 



Another form, to render its meaning plainer. Let X^, /i-i, v, 

 and X25 A'2j ^2 be the direction-cosines of the elements referred 

 to rectangular axes, with the .2;-axis, to which Xi and Xg refer, 

 chosen as the line joining the elements. Then* 



^^mmi!^i^\\,+i.,,.,+v,v,). . . (8) 



J. J. Thomson^s estimate is f 



M=i/*oM2^i^2-^ (^) 



Comparing this with (8) we see that there is a notable 

 difference. 



7. The mutual energy being different, the forces on the 

 charges, as derived by J. J. Thomson by the use of La- 

 grange^s equations, will be different. When the speeds are 

 constant, we shall have simply the before-described vector 

 product (4) for the " electromagnetic force ;" or 



Fi=/-to^iVuiH2, F2 = /^og2VTi2Hi . . . (10) 



if Fi is the electromagnetic force on the first and F2 that on 

 the second element, whilst Hi and H2 are the magnetic forces. 

 Similar changes are needed in the other parts of the complete 

 mechanical forces. 



It may be remarked that (if my calculations are correct) 

 equation (7) or its equivalents expresses the mutual energy of 

 any two rational current- elements [see § 1] in a medium of 

 uniform inductivity, of moments qiu^ and $'2^2? whether the 

 currents be of displacement, or conduction, or convection, or 

 all mixed, it being in fact the mutual energy of a pair of 

 definite magnetic fields. But, since the hypothesis of instan- 

 taneous action is expressly involved in the above, the applica- 

 tion of (7) is of a limited nature. 



8. Now leaving behind altogether the subject of current- 

 elements, in the investigation of which one is liable to be led 

 away from physical considerations and become involved in 

 mere exercises in diflFerential coefficients, and coming to the 

 question of the electroaiagnetic effects of a charge moving in 

 any way, I have been agreeably surprised to find that my 

 solution in the case of steady rectilinear motion, originally an 

 infinite series of corrections, easily reduces to a very simple 

 and interesting finite form, provided u be not greater than v. 

 Only when u>v is there any difficulty. We must first settle 



* ' Electrician,' Jan 24, 1885, p. 221. 



t ' Applications of Dynamics to Physics and Chemistry,' chap. iv. ; and 

 PhU. Mag. AprU 1881. 



