Mechanical Theory of Circular and Elliptic Respiration. 405 
each molecule describes an ellipse, the axes of which are parallel to 
those of x and y; that is to say, if we make 
£ = pcosg, 7 = 7sing, 
2 
Acai hint 
ue CB 
and consequently 
A£=p(sin26 sing.—2 sin’ 4 cos ¢), 
Ay = — q(sin24 cos ¢ + 2 sin’ 6 sin ¢), 
where 6 = a , we shall find, by substituting these values in the 
equations (2.), which must hold good independently of $, 
s=A+Ckh &=B — ae 
Sfsin26—2kEhsin°G=0, + ------ (4.) 
zg sin 204 — Ehsin’d=0, J 
wherein & = 1 expresses the ratio of the semiaxes of the elliptic 
vibration, and 
A' = 
Ms fsin?f Bla ~ 29 
or f sin B = 5 
U 7 _ 
0} = 7,2 2h sin 26. 
Equating the two values of s*, we get, for the determination of k, 
the following quadratic :— 
A'—B' 
ke + Gi hich ees Rue ee Bs (5.) 
= ah making the substitutions (3.) in equations (1.), page 400, we 
ave 
Undulatory Theory, as applied to the Dispersion of Light and other Sub- 
jects,” has dwelt at great length on the theory of elliptic polarization, 
which they have been supposed to afford, and which he regards as a most 
important accession to the science of Light. Professor Powell has also 
made some communications on the subject to the British Association, and 
has written two papers about it in the Philosophical Transactions (1838, 
p. 253 ; and 1840, p. 157), besides several others in the Philosophical Ma- 
gazine. He, however, always attributed this theory of elliptic polarization 
to Mr. Tovey, until his attention was directed, by a letter from M. Cauchy, 
to some investigations of the latter which he had not previously seen (Phil. 
Mag. S.3.vol.xix. p.374). Mr. Tovey set out with the principles ot M. Cauchy, 
and therefore naturally struck into the same track, in pursuit of the same 
object, apparently quite unconscious that any one had preceded him. It 
was, indeed, an obvious reflection, that these principles, when generalized 
to the utmost, ought to include, not only the laws of elliptic polarization, 
but (as really has been thought by M. Cauchy and his followers) of disper- 
sion and absorption, and, in short, of all the phaenomena of optics, 
