of Edinburgh, Session 1873-74. 223 
Here it is to be observed that Ampere’s directrice for the circuit 
cq is 
the integral extending round the circuit; so that, finally, 
<pa — - rSoqVVa'0 . 
4. We may obtain from the general expression above the abso- 
lutely symmetrical form, 
rV. a/acq 
TW* ’ 
if we assume 
/( Ta) = const , F (Ta) = ~ . 
Here the action of a' on a x is parallel and equal to that of cq on a'. 
The forces, in fact, form a couple, for a is to be taken negatively 
for the second — and their common direction is the vector drawn to 
the corner a of a spherical triangle a be, whose sides ab, be, ca in order 
are bisected by the extremities of the vectors Ua', Ua, Ucq. Com- 
pare Hamilton’s Lectures on Quaternions , §§ 223-227. 
5. To obtain Ampere’s form for the effect of one element on 
another write, in the general formula above, 
/(Ta) = ^ , F(Ta) = 0 , 
and we have 
1 . , o « F aSaa'~l V.a'Vaa, 
avJ T — ’ 
_ _ c^Saa' aSctja' SaSaa/Sacq ( V.a'Vaa, 
~Ta? TV~ ~ Ta 5 + ’ 
== + ~ Saa'Saa^ , 
= — ,YaaYaa x -f ^ Saa'Saa^ , 
which are the usual forms. 
6. The remainder of the expression, containing the arbitrary 
terms, is of course still of the form 
- S( ai V) [aS(a'V)/(Ta) + a'F(Ta)] . 
2 f 
VOL. VIII. 
