3 a6 PETER GUTHRIE TAIT 



congruent to the corresponding one in the former figure (when the two figures are 

 drawn to the same scale), but pushed farther to the right as its extremities are less 

 inclined to the horizon. In its new form the curve of vertices ABC leans back 

 towards the eye, and we have an inverted image. The lower medium need not be 

 uniform as, for simplicity, we assumed above. All that is required is that the rate 

 of diminution of density upwards shall be less in it than in the upper medium. 



Those who have followed me so far will at once see that, as a more rapid 

 increase of density, commencing at a certain elevation, makes the curve of vertices 

 lean back, so a less rapid decrease (tending to a " stationary state ") at a still higher 

 elevation will make the curve of vertices again lean forward from the eye. I need 

 not enlarge upon this. 



Thus to repeat : the conditions requisite for the production of Vince's pheno- 

 menon, at least in the way conjectured by him, are, a stratum in which the refractive 

 index diminishes upwards to a nearly stationary state, and below it a stratum in 

 which the upward diminution is either less or vanishes altogether. The former con- 

 dition secures the upper erect image, the latter the inverted image and the lower 

 direct image. 



In my paper read to the Royal Society of Edinburgh I have given the mathe- 

 matical details following from the above statement ; and have made full calculations 

 for the effect of a transition stratum, such as must occur between two uniform strata 

 of air of which the upper has the higher temperature. From Scoresby's remarks it 

 appears almost certain that something like this was the state of affairs when the 

 majority (at least) of his observations were made. When two masses of the same 

 fluid, at different temperatures, rest in contact ; or when two fluids of different 

 refractive index, as brine and pure water, diffuse into one another; the intervening 

 layer must have a practically " stationary " refractive index at each of its bounding 

 surfaces, and a stratum of greatest rate of change of index about midway between 

 them. The exact law of change in the stratum is a matter of comparatively little 

 consequence. I have assumed (after several trials) a simple harmonic law for the 

 change of the square of the refractive index within the stratum. This satisfies all 

 the above conditions, and thus cannot in any case be very far from the truth. But 

 its special merit, and for my purpose this was invaluable, is that it leads to results 

 which involve expressions easily calculated numerically by means of Legendre's 

 Tables of Elliptic Integrals. This numerical work can be done once for all, and 

 then we can introduce at leisure the most probable hypotheses as to the thickness of 

 the transition stratum, the height of its lower surface above the ground, and the 

 whole change of temperature in passing through it. I need not now give the 

 details for more than one case, and I shall therefore select that of a transition 

 stratum 50 feet thick, and commencing 50 feet above the ground. From the 

 physical properties of air, and the observed fact that the utmost angular elevation 

 of the observed images is not much more than a quarter of a degree, we find that 

 the upper uniform layer of air must under the conditions assigned be about 70 C. 

 warmer than the lower. Hence by the assumed law in the stratum, the maximum rise 

 of temperature per foot of ascent (about the middle of the transition stratum) must 

 be about o "2 C. per foot. Such changes have actually been observed by Glaisher 



