On Atmospherical Refraction. 355 



instead of retracing the steps of the calculation with these cor- 

 rections only, it will be more satisfactory to extend the general 

 theorem somewhat further, without confining it to a particular 

 law of temperature. 



5. For this purpose we may make -^ ;^ ^, _i ;— ^' — —^z. t' 



dz dr dr 



and we shall have, for computing the coefficients of the series 



Av rr 

 vr + •- + . . ., the values 



V 



du 



dr 



dx fu 



dr "~ mpsz 



dz — dM — V 



dr ^sdr ps ' 



dy ^dz — ^v 



dr dr ps 



dr ^x 



dr mpsz 



, dv ,^ X , t , tx , tx 



d :i-==d^. + dx. — dz.^ dz -2_ vdr; and 



dr mpsz mpsz mpsz mpsz' 



ddu t'x t'v txv XT • • , „o„ 



-,— :; ==— 1 — , , ., - + , ■ V. Now smce m is about 766 



dr" mpsz m'p^S'z mp s z- ' 



it is obvious that the second term, containing its square, may be 

 neglected in comparison with the third, since the other quan- 

 tities concerned in these terms can never differ materially from 

 each other : for the same reason the term v may be omitted, as 

 not being divided by p, and u may be considered as equal to s, 

 and its fluxion neglected, as well as that of x, which may be 

 called = 1 ; and we may proceed to take the fluxion of 



ddu t'x txv t' tv 1 ji ddt> 



~r 2 'i^a ""■ I 



whence d 



dr^ mpsz mph'z" mpsz mp''s'z^ dr^ 



d^' ^dz t^v + vdt 2^dz , d'w ^" 



4- -^ ■ ^ 2. ; and -— , = -5 f. 



mpsz mpsz' nip's^z' mp's'z^ dr' mpsz 



mp's'z' my^s z ■ \mpsz / mp^sz' mp^s'z 



