648 Mr W. HOPKINS, ON THE EXTERNAL TEMPERATURE OF THE EARTH, 

 Earth's surface and the atmospheric particles within the sphere of this mutual radiation. On this 



supposition, which may be readily admitted, — will be small, and we shall have approximately 



X 



„ „ KG COS 9 



Poisson shews how the complete value of £ might be determined, but, as already 

 intimated, experiment altogether fails to give us the data which are required to render this 

 theoretical determination practically available. We have much better means, however, of 

 determining the approximate value of the second term in the above expression for £. 



If r denote the instantaneous distance of the Earth from the Sun, we shall have 



s 



a = 1' 

 r 



where S is a measure of the intensity of the Sun's radiation. Also 9 is the Sun's zenith dis- 

 tance, and, therefore, 



cos 9 = sin n sin y sin v + cos ix cos \|/ \/l — sin 2 y sin* v, 

 where /u = latitude of the place, 



y = obliquity of the ecliptic, 

 v = Sun's longitude, 



\J/ = Sun's angular distance from the meridian, v and \^ being 

 consequently functions of the time t. If then we substitute for cos 9, we have 



KG COS 9 kS 



X Xr 



\ sin m sin y sin v + cos fx cos \J/ \/\ - sin 8 y UD*yi, 



in which expression, neglecting the eccentricity of the Earth's orbit, r may be taken as the 

 mean distance of the Earth from the Sun. The expression will only be true when the Sun 

 is above the horizon of the place, a cos 9 being zero when the Sun is below the horizon. 



Hence, — is a discontinuous function of \f/, and can be expressed by a well-known 



\ 



formula in a series of terms containing respectively the cosines of \|/ and of its multiples. 



Effecting this transformation, our author obtains 



kg cos 9 kS . Tr Tr . _ „ , 

 = ) V + V 1 cos vl/ + V 2 cos 2 yL + &c i , 



where 



V = — sin ^ sin 7 sin v + Q + Q, cos 2u + Q 2 cos iu, &c, 



2 



Q, Q,, &c. being numerical quantities the values of which are immediately expressible by 

 means of elliptic integrals. P,, Fj, &c. may be expressed in a similar manner. Thus 

 we have 



^ + ^ (F+FlC0Sv/ ' + &C - ) ' 



