26 Law of the Induction of an Electric Current upon itself. 
member of the last equation shall be ee of z or of 9”. 
We shall first show that this requires that 
sane 
7 Fdr 
-R sin 9’ 
be also independent of z. It is obvious that the value of this 
integral is determined by that of sin 9’,* so that 
+R sin gq’ 
es Fd z=f sin 9’ 
-R sin 9’ 
which being substituted in (F’) we have 
“is 
J feng de 
for the integral part whose variation consequent upon a change 
in the value of z must always be zero. Now we have, dp 
being =, ar R cos 9’ =z cos g, 
Ph ele See d¢ 
t Smee! ene ‘gi i£ 
. = ft In 
df sin g/ 
dsing’’ 
"dg, 
where /’ sing’ is put for 
whence we must have 
= cos? gq’ 
er fies d — 0 
for all values of = from Oto R. But it is obvious that < 
is necessarily positive throughout the integration, and hoetore 
complete integral may be zero, f’ sin g’ must 
either be always zero or else must change its sign between the — 
limits of integration, and that for all the values of z. Now this | 
last is impossible, for whatever be the nature of the function, 
J’ sin g’, its value and sign must be the same above the limit of 
g’ =4 7 as below it,} and limits $7—v and 47+» can be assumed 
for the values of ¢’, within which the value of f’ sin ¢’ being, if | 
possible, greater than zero, does not change its sign. But the 
value of z can be taken so small as to bring the values of 9 
within those limits throughout the whole sd Sis The : 
* ae that this is necessarily true, but that it is so in the prostet case where the ; 
of F is the same whet ’ be greate r or less than 
a necessarily but because this ia: the case with the fenetions Sf sing 
