of Edinburgh, Session 1873-74. 225 
8. The vector couple exerted by a L on a! must obviously be ex- 
pressible in the form 
V.a Va, , 
where w is a new linear and vector function depending on a alone. 
Hence its most general form is 
-na x — Poq + QaSacq , 
where P and Q are functions of Ta only. The form of these func- 
tions, whether in the expression for the force or for the couple, 
depends on the special data for each particular case. Symmetry 
shows that there is no term such as 
RVaoq . 
9. As an example, let cq and a be elements of solenoids or of 
uniformly and linearly magnetised wires, it is obvious that, as a 
closed solenoid or ring-magnet exerts no external action, 
(pa = — ScqV. \f/a ' . 
Thus we have introduced a different datum in place of Ampere’s 
No. III. But in the case of solenoids the Third Law of Newton 
holds — hence 
<pa' = Sc^VSaV.^a , 
where x is a linear and vector function, and can therefore be of no 
other form than 
af (To). 
Now two solenoids, each extended to infinity in one direction, act 
on one another like two magnetic poles, so that (this being our 
equivalent for Ampere’s datum No. IV.) 
Hence the vector force exerted by one small magnet on another is 
pS^VSa'V^. 
10. For the couple exerted by one element of a solenoid, or of 
a uniformly and longitudinally magnetised wire, on another, we have 
of course the expression 
V. a' zzra x , 
where w is some linear and vector function. 
