130 REPORT—1 885. 
but the number of lines of force passing through the circuit 
=|{cu + mb + nc)d8, 
where a, b, c are the components of magnetic induction, so that 
= 7H_aG 
dy dz’ 
pate _ do - 
dz dx’ 
dG _d¥F 
To connect a, b, c with the current, Maxwell makes use of the prin- 
ciple that the line integral of the magnetic force taken round any closed 
curve equals the current flowing through the curve. This leads to the 
equations— 
dy dp 
try Oe 
dy dz’ 
la d 
Ary" —&Y 
dz dx’ 
4. wt da. 
da dy’ 
so that if u be the coefficient of magnetic permeability, 
Ae scape tee, 
yi dy dz 
and so on. Substituting the values of a, b, c, given above, we find 
g g 
d fd¥F , dG , dH 
4 a { a ae bees \ —v7F 
iets da | da: Cay aw dz aia 
with similar equations for G and H. 
Now v. Helmholtz, in his paper ‘ Ueber die Bewegungsgleichungen der 
Elektricitit fiir ruhende leitende Koérper’ (Crelle, Ixxii. p. 57 ; Gesam- 
melte Werke, ii. p. 545), has investigated the most general expressions 
for F, G, H, consistent with the force between two closed circuits agree- 
ing with that indicated by Ampére’s theory, and he finds that if the 
circuits are closed circuits, as Maxwell assumes all circuits to be, then 
ae eh 
de dy dz ’ 
and therefore Arpu=— 7?F, 
with similar equations for Gand H. These equations are sufficient to 
determine the quantities F, G, H. 
Maxwell does not at once put dF/dz+dG/dy+dH/dz=0; he writes J 
for this quantity, and puts 
x 
Then Bal [(M de dy de +X; 
