Sir W. R. Hamilton on a Lens qfUniaxal Crystal. 289 



Now <p = co + $ — co, hence 



cos $ = cos co cos (<$ — «) — sin co sin (<f> — co), or 



cos 2 <p = cos 2 co cos 2 (4> — co) 4- sin 2 co sin 2 (<$— co) — 2 sin co cos co 



cos ($—co) sin (<|>—co). Similarly, 



sin 2 <p = sin 2 to cos 2 ($ — so) +cos 2 co sin 2 ($— co) + 2 sin co cos co 

 cos ($— co) sin (<p — co). 



Putting these values of cos 2 <p, sin 2 <$ in the first member of 

 (V.) and eliminating, we find 



(A-B) sin 2 co sin 2 (<p— co) = . . . . (VI.) 



Now as $ is quite arbitrary, we may always assume it dif- 

 fering in value from co, hence we may consider sin 2 ($ — co) 

 as always different from zero ; dividing then by this factor, 

 equation (VI.) is thus reduced to 



(A-B) sin 2 co = (VII.) 



This equation is satisfied when A = B by any values of co, 

 or, in other words, when the moments of inertia round two of 

 the principal axes passing through the centre of gravity are 

 equal ; any pair of rectangular axes in a plane perpendicular 

 to the third principal axe are also principal axes. 



When A is not equal to B, equation (VII.) can only be 



satisfied by the values co = 0, or co = — -, or the principal 



axe O x' coincides with O x, which is parallel to G X, hence 

 we may deduce the following theorem : — 



The principal axes of rotation of a body passing through 

 any point assumed on one of the three principal axes through 

 the centre of gravity, are always parallel to these axes. 



XLII. On the Focal Lengths and Aberrations of a thin Lens 

 of Uniaxal Crystal, bounded by Surfaces which are of Revo- 

 lution about its Axis. By Sir William Rowan Hamil- 

 ton, P.R.I. A., Member of several Scientific Societies at 

 Home and Abroad, Professor of Astronomy in the University 

 of Dublin, and Royal Astronomer of Ireland*. 



r T 1 HE following short investigation may perhaps be not 

 A without interest to the students of mathematical optics, 

 as serving to illustrate a general method, and to correct an 

 important error into which an eminent writer has fallen. 



1. Let a ray of ordinary light, in vacuo, and in the plane 

 of xx, proceeding from or towards a given point on the axis 

 of z, be incident nearly perpendicularly on a given surface of 

 revolution, and there undergo extraordinary refraction at en- 



* Communicated by the Author. 



Phil. Mag. S. 3. Vol. 19. No. 121. Oct. 1841. U 



