Galvanism, from Galvani to Ohm. 91 



contains ds and ds' linearly and homogeneously, as it should. 

 We can also add any terms of the form 



rf{r..(ds'.r). x (r)|, 



where \(r] denotes any arbitrary function of r, and d denotes 

 differentiation along the arc s, keeping ds' fixed (so that 

 dr = - ds) ; this differential may be written 



- ds . (ds'. r) . x (r) - r x (r) (ds'. ds) - * x '( r ) r (ds . r) (ds'. r). 



In order that the law of Action and Eeaction may not be 

 violated, we must combine this with the former additional term 

 so as to obtain an expression symmetrical in ds and ds' : and 

 hence we see finally that the general value of F is given by the 

 equation 



F = -n'rj j |(ds.ds')-J(ds.r)(ds.r)j 



+ x (-; (ds' . r) ds + x( r ) (ds . r) . ds' + x (r) (ds . ds')r 



+ i x '(r)(ds.r)(ds'.r)r. 

 The simplest form of this expression is obtained by taking 



when we obtain 



/ 

 F = - {(ds . r) . ds' + (ds'. r)ds - (ds . ds')r} . 



The comparatively simple expression in brackets is the 

 vector part of the quaternion product of the three vectors 

 ds, r, ds'.* 



From any of these values of F we can find the ponderomotive 

 force exerted by the whole circuit s on the element ds' : it is, in 

 fact, from the last expression, 



u'f 1 



[ ? - 3 ((ds'.r).ds-(ds.ds>}, 



* The simpler form of F given in the text is, if the term in da' be omitted, the 

 form given by Grassmann, Ann. d. Phys. Ixiv (1845), p. 1. For further work on 

 this subject cf. Tait, Proc. R. S. Edin. viii (1873), p. 220, and Korteweg, Journal 

 fiir Math, xc (1881), p. 45. 



