92 
MR. J. G, LEATHBM ON THE THEORY OP THE 
The restriction of the relation to this form is justified as follows : — 
When there is no matter present the polai’isation {f', g', li) is null, and there is no 
rotatory effect. When there is matter present the action of an electric field (P, Q, P) 
on the polarisation induced by it gives rise to a pondero-motive force which does 
mechanical work in a small displacement of the matter, equal, for an element of 
volume St, to 
(P Sf' -hQSg' + 11 S/i) St. 
When there is no conduction and therefore no dissi])ation, this quantity must be 
an exact diffei’ential, otherwise mechanical work could be gained in a complete cycle 
of displacement of the material medium, which would imply the possibility of 
perpetual motions. This restriction must also hold universally, because the nature of 
tlie molecular polarisation is independent of whether conduction is present or not. 
It requires that j (P Sj" + Q Sg' + P Sh') St and therefore | (/"' SP SQ + h' 8P) cIt 
shall be the variation of | F cIt, where F is some function of (P, Q, P) and its 
dilferential coefficients, terms at the time limits being left out of account. The 
expression for ( f, g', h') in terms of {P, Q, P) can therefore involve no rotatory 
terms in (P, Q, P) itself, as Lord Kelvin first shewed, but it may have rotatory 
terms in d/dt (P, Q, Pt), which have the characteristics of the magneto-optic property. 
Potatory terms of a certain type in tlie spacial fluxions of (P, Q, P) are also 
admissible ; these lead to optical rotation of the structural kind ; they are foreign to 
the present problem because they are isotropic, instead of being related to an imposed 
vector, the Intensity of magnetisation. In either case higher differentiations of odd 
order might also come into the expressions : these would affect the relations of the 
phenomena to optical dispersion, but not the questions here treated. 
Eguatious of Pro^Ktgatio^i. 
G From the fundamental equations we readily obtain 
u — 
V = 
w + (''a i Q 
-zy = CT fi- 
Airid dt 
K gf 
TttC^ dt 
/ d- \ 
dd 
^ + f/s) P “ 
^ + 7372 .7/ ^ Q + ,7f2 + ~ (h .7/3 + P'3 ) P 1* 
df- 
For brevity we may put 
Pi. V- 2 > Vs 
in 
( 8 ) 
