August 22, 1895] 



NA TURE 



403 



resulting value for Mars and the known decrease in illumination 

 due to the telescopic maj;nification employed. 



To deduce the resulting irradiation we must find the value of 

 which renders the above equation a maximum, and then substi- 

 tute this value in the equation. To do so directly leads to an 

 equation of so high an order that approximation will be found 

 the better, if indeed it lie not the only, method of solution. By 

 this means it appears that the necessary correction does not be- 

 come insensible, to three places of decimals, till the phase angle, 

 7, somewhat exceeds 30". 



The formula must be used within the limits for which 



sin a 1 1 .u sin a 



- I ; beyond them 



must lie taken as 



sin(7-t-o) ' J -■ " sin (7-1- a) 



unity. 



If the reflection from the disc followed the law of the cosines — 

 that is, if the apparent illumination were always equal to the 

 tnie one — we should have 



'/( sin o W 



cos 7 



COSMO 



7)) 



where a, 7, and >i have their previous values, and w = a constant 

 to be determined from the ei)uation, from the value at the limb. 



But although this is the formula for the case of a theoretical 

 rfiugh bare globe, it manifestly does not hold in the case of Mars, 

 of « hich the limbs are not only as bright as the centre of the disc, 

 but much brighter. The previous formula is, therefore, to be pre- 

 ferred to it, although even that formula makes the irradiation 

 correction at the terminator too great as compared with that at 

 the limb. 



But it is to be specially noticed that no law of correction for 

 irradiation at the terminator, however big it make that correc- 

 tion to be, is able to do away with the outstanding differences, 

 presently to be noted, of the equatorial diameter at different 

 times upon which the evidence of the twilight arc is based. 



There is also the correction for phase. Inasmuch as tlie 

 phase axis and the polar axis did not in general coincide, there 

 entered into its determination beside the amount of the lacking 

 lune, the angle of inclination of the two axes. So that the 

 amount of the defalcation had to be calculated in accordance for 

 each night. These corrections and their results reduced to 

 distance unity have been calculated and tabulated. 



Besides the above there is a fifth correction neetled to reduce 

 the diameter measured tor the polar one, to the true polar (iia- 

 meter. The diameter measured perpendicular to this, or the 

 apparent equatorial diameter, although not in fact an equatorial 

 diameter, was always exactly equivalent to one, since its ex- 

 tremities were always each 90' distant from the pole. The other, 

 however, was the diameter of the ellipse made by the plane 

 passing through the polar axis, w hich was inclined to the polar 

 axis by the angle of lilt, and needetl, therefore, to lie re^luced to 

 that ellipse's minor axis. This correction is best applied to the 

 means, and appears in the subjoined table. 



Polar Diameters. 



As previously explained, no correction is needed for astigma- 

 tism, as the measures themselves correct it. 



So soon as the measures had been corrected and reiluced to 

 distance unity, two things became apparent, both so large as to 

 be almost unmistakable licfore taking the means. The first was 

 the jxilar flattening ; the other an equally systematic difierence 

 in the size of the equatorial diameter according as the measures 



NO. 1347, VOL. 52] 



were made in October or in November. The November mea- 

 sures came out much larger than the October ones ; while the 

 corresponding polar measures, on the other hand, showed no 

 corresponding increase. Struck by this fact, and suspecting its 

 cause, instead of taking thei mean of all the mea.sures for each 

 diameter, I divided them into sets according to their proximity 

 in date to the time of opposition, and took the mean of these 

 sets. 



The means are as follows : — 

 I Polar Diaiueter. 



Mean October 15 to October 23, both dates inc. 9 '379 



12 „ 30, „ ,, 9J-378 



,, Nov. 2 to Nov. 21, ,, „ 9"'390 



Equatorial Diameter. 

 Mean October 15 to October 23, both dates inc. 



,. >, 12 ,, 30 .. .. 



,, Nov. 2 to Nov. 21, ,, ,, 



9 420 

 9' 440 

 9"-545 



Opposition occurred on October 20. The first set in each sche- 

 dule, therefore, was made within four days of opposition ; the 

 second, within eleven days of it; the last, from fourteentothirty-two 

 days after it. That there is a systematic increase in the equatorial 

 measures is apparent. That it is not paralleled by a correspond- 

 ing increase in the polar ones shows instantly that it can hardly 

 have been due to systematic error in the observer, since in that 

 case both sets of measures should, in all probability, have been 

 affected. 



Now as all the measures had previously been corrected for re- 

 fraction, irradiation, phase and tilt, the means of each diameter 

 should have agreed with themselves. The polar did so in a very 

 satisfactory manner ; the equatorial not only did not, but they 

 differed in proportion to their distance in time from the date of 

 opposition. Now the only factor that increased in proportion to 

 the distance in time from opposition was the phase. The 

 direct effect in the way of decreasing the equatorial diameter had 

 already, as we have seen, been allowed for ; what is more, it is 

 a correction susceptible of great accuracy, since it depends upon 

 the motions and relative distances of the earth and Mars, 

 quantities very accurately known. Besides these quantities, 

 there is nothing which enters into the calculation but the 

 position of the pole of Mars, and this would have to be, not only 

 some 35 Martian degrees in error to explain the discrepancy, but 

 would have had to have shifted obligingly to an opposite error 

 during July and August to account for the measures taken then, 

 as we shall see later. In other words, no such discrepancy exists. 

 In the case of a bare globe this direct effect would be the only 

 eflTect phase could have upon the equatorial diameter ; not so, 

 however, in the case of a body not bare. If a planet possessed 

 an atmosphere, that atmosphere would cause the phenomenon of 

 twilight, and to an observer at a distance the effect of the 

 twilight would be to prolong the terminator beyond what would 

 otherwise be its limits. There would thus result a seeming 

 increase in the equatorial diameter as the disc passed frcjm the 

 full to the gibbous phase. Now this increase is precisely the 

 increase that the measures disclose. 



It is furthermore worth noting that in the absence of an 

 atmosphere, the measures of the equatorial diameter as the 

 phase increased would not only have shown no increase, but 

 would actually have shown a decrease, inasmuch as it would be 

 impossible for an observer to see quite out to the edge under the 

 diminishing illumination. 



To determine the extent of the twilight thus disclosed by the 

 measures, the angle I)etween the radius to the sunset point and 

 the radius prolonged to the point of the atmosphere last 

 illuminated, had to be found. This enabled an equation to be 

 developed, which gave for the visible twilight fringe an arc of 

 5', the double of which, or 10°, is the angle which determines the 

 duration of the twilight, or the twilight arc. On the earth this 

 angle is 18°. 



Applying the correction due to the twilight fringe, to the 

 means previously obtained, we find the following close agreement 

 between them : — 



Polar Diameter. 



October 15 to 22 inc 9 "'356 



October 12 to 30 ,, 9"'354 



November 2 to 21 ,, 9' '353 



Equatorial Diameter. 

 October 15 to 23 inc. ... ... 9 ""404 



October 12 to 30 ,, ■ 9"'396 



November 2 to 21 ,, 9" -402 



