294 



Lord Kelvin 



on 



sec/3 (cos 2 # + 1) taken to make it equal to Majorana's r for sun's 

 zenith distance 44°*6, on the supposition that the region of 

 sky observed was in each case (see § 62 above) in the position 

 of minimum luminosity as given by (19). It is obvious that 

 this position is in a vertical great circle through the sun, and 



Col. 1. 



Col. 2. 



Col. 3. 



Col. 4. 



Col. 5. 



Time. 



Zenith 

 distance 

 of sun. 



1- 



Ratio of 



luminosity of 



sun's disc to 



luminosity of 



sky. 



r. 



8 



s' 



Zenith distance 

 of least lumi- 

 nous part of 

 sirs-. 

 (5. 



5.50 a.m. 



. 7 

 . 8 



1 9 

 11 



81-7 

 68-0 

 56-1 

 446 

 29-9 



2570000 

 3125000 

 3650000 

 3930000 

 3760000 



3280000 

 3350000 

 3600000 

 3930000 

 4600000 



5% 

 14-4 

 217 

 27-8 

 33-6 



on the opposite side of the zenith from the sun ; and thus we 

 have 0=?-r-/3. Hence (19) becomes 



|=ggs[ r( ^~ J) ]Wec/3[cor(g+£)+11 . (20)/ 

 To make (20) a minimum we have 



The value of /3 satisfying this equation for any given value 

 of £ is easily found by trial and error, guided by a short 

 preliminary table of values of /3 for assumed values of /? + J. 

 Col. 5 shows values of thus found approximately enough to 

 o-ive the values of S/s shown in col. 4 for the several values 



of r. 



§ 70. Confining our attention now to Majorana's obser- 

 vations at 9 a.m. when the sun's altitude was about 44 c *6: 

 let e be the proportion of the light illuminating the air over 

 the crater of Etna which at that hour was due to air, earth, 

 and water below; and therefore 1 — e the proportion of the 

 observed luminosity of the sky which was due to the direct 

 rays of the sun, and expressed by § 68 (19). Thus, for @= 27 c *8, 

 ?=44°-6, and 0=72°'4, we have S/s = 3930000/ (1-e), instead 



