On the Geodesic Lines on an Oblate Spheroid. 329 



cotton-wool. When cold the tubes were put into a freezing-mix- 

 ture at 10° F., and afterwards into one at 0° F., apparently with- 

 out any effect. The tubes were set aside with the cotton- wool 

 undisturbed, and they remained at rest during a week. On again 

 examining them the cotton-wool was removed ; but there was no 

 sign of crystallization until, on closing one of the tubes with the 

 thumb and inverting it, a large mass of crystals became visible 

 in consequence of the draining off of the mother-liquor, now only 

 a saturated solution. Air entered some cavities in the crystals, 

 and on turning back the tube so as to allow the mother-liquor 

 once more to envelope the crystals, these air-filled cavities stood 

 out with most perfect definition, while the crystals themselves 

 again became invisible. This experiment very much impressed 

 me with the value of Sir David Brewster's suggestions ; and I 

 cannot fancy a better test in the hands of an intelligent lapidary 

 for detecting flaws and cavities in precious stones before deciding 

 on their value or proceeding to cut and polish them. 



On repeating the experiment with the double salt, I found that 

 at about 0° F. the solution became solid, but so transparent that 

 no casual observer would suspect it to be so. One of the tubes 

 that was more than three-fourths full had a few scattered needles 

 at the surface, showing that crystallization had set in. On pass- 

 ing down a platinum spatula the solution was found to be pulpy, 

 so much so that the tube could be inverted without any loss of 

 liquid. By repose the pulpy salt became crystalline, and the 

 mother-liquor, of the same refractive density, separated. 



This sodio-zincic sulphate, as obtained in an open evaporating- 

 dish, contains only four equivalents of water. In a closed tube, 

 if left to repose during some weeks, it assumes a different state 

 of hydration ; and as it does so it acquires a different index of 

 refraction as compared with that of the mother-liquor, and so 

 becomes visible. 



Highgate, N., 

 September 8, 18/0. 



XLIII. On the Geodesic Lines on an Oblate Spheroid. 

 By Professor Cayley, F.R.S* 



THE theory of the geodesic lines on an oblate spheroid of 

 any excentricity whatever was investigated by Legendref ; 

 and the general course of them is well known, viz. each geodesic 

 line undulates between two parallels equidistant from the equator 

 (being thus either a closed curve, or a curve of indefinite length, 



* Communicated by the Author. 



t Mem. de VInst. 1806; see also the Exer. de Calcul Integral, vol. i.(181 1) 

 p. 178, and the Traite des Fonctions Elliptiques, vol. i. (1825), p. 360. 

 Phil. Mag. S. 4. Vol. 40. No. 268. Nov. 1870. Z 



