110 G. F. Becker — Certain Astronomical 



Let the amount of sunshine received upon the unit area at 

 the unit distance from the sun in the unit of time be H. Then 

 in an iustant of time the amount received upon the zone of 

 unit width at a distance r from the sun would be 



-dW H , 

 at. 



dX ' r~ 



By the principle of the conservation of the moment of mo- 

 menta, the radius vector of an unperturbed planet sweeps over 

 equal areas in equal times. If h is a constant this, which is 

 one of Kepler's laws, is expressed by i ? dd- = hdt. By substi- 

 tution in the last expression this gives the following measure 

 of the receipt of solar energy for a small change in the earth's 

 longitude, 



dW H 7r . Ha' sin 2 A . , . , _ . . . _ „ .. ., _ , . 



=--rd$= -—, cotAJ(3) — sine sin B.XldB (1) 



dX h h v . 



and the whole heat received between the autumnal and the 

 vernal equinoxes on a zone of unit width will be proportional 

 to the integral of this quantity from d- = to # = n, or to 

 twice the integral from zero to tt/2, since the conditions are 

 symmetrical. To find the corresponding value for the summer 

 interval it is only necessary to substitute tt-X for X. 

 To facilitate integration it may be noted that 



sin Sr.X.dS= cos B dX-d(X . cos 3) 



and 



dX sin s cos B tan A 



~dB ~ ~ (l-sin 2 £sin 2 3)z7($) 



These values reduce (1) to 



c?WH „ Ha- 2 . •, ( , ,.., 



pr -rd$ — —r- sin 2A^ cot A /1(B) + 



dX h h { 



tan X (m~ (1-smCJ 3)^(3) ) + ^( X - C ° S5 Sinf ) [**' (2) 



For values of X from to tt/2— e, or from the equator to the 

 polar circle, this is integrable term by term. If sin e/cosA = x 

 and if E^x), F^x), H\x) denote complete elliptic integrals of 

 the three classes for the modulus x ; and if Z is the integral of 

 (2), being the solar radiation received between equinoxes on 

 the zone of unit width, 



2Ha" \ 



Z='-^- sin 2A -j cot A E^x) + tan AfF 1 ^) - cos's II 1 (h)] 



± \ sin e J- (3) 



