26 INTENSITY OF SUN'S HEAT AND LIGHT. 



for sin H cos D its equal from (21), and for cos 2 D its equal 1 — sin 2 o sin 2 T; then 

 multiplying by cos T, we have, 



/„-, , ^ C tan L sin a cos 2 T d T 

 cosTd H = J . -- 



(1 — sin 2 o sin 2 T) J 1 — (^J sin 2 T 



Here in a changed form, sin a cos 2 T — sin a — sin a sin 2 T, which is equal to 



— — |~(T — sin 2 o sin' 2 T) + sin 2 a — 11; therefore writing — cos 2 a in place of sin 2 co — 1, 

 sin o LV ' 



and then separating the expression into two parts, we obtain after cancelling the 



common factor, 



rtanL\ AT cos 2 to (IT ) ,. . . 



CcosTdH= I -- — t — === , ., .--= == . (24.) 



l rp 



Now let J — -. — - , -= = F, an elliptic function of the first species ; and 



J , (sin to\ 2 . z m 



< l -\c^L) SmT 



/d T 

 — — s r-TTTvyr— t— — = = II, an elliptic function of the third species, 



( 1 -' wow, ^Ji-(^J)'«»-r 



according to Legendre and other geometers. 



Adopting these designations, we have now defined the terms of Cu d x. Passing 



over \ u, as already known, the next term of the general formula of summation (20) , 



is T L _ ; which is determined as follows : Taking the logarithmic differential of (13) 

 d x 



in its simplified form, 



u = A 2 sin L sin a sin T (H — tan H). 



du 2d A m7rT7 / 1 \ dH 



— " — J— + cotTdT+ll— — TTrlrf ; IT- 



u A \ cos'H/H — tan H 



Again, taking the logarithmic differential of the first and last members of (22), 



2d A —2esin6dT 

 recollecting that d 6 = d T, we find — -r— = — t— « — • Also equating the 



1 A 2 n V 1 e 2 



first and third members of (22), -5— = -j-rp . And the value of d H has 



already been found; whence by making the indicated substitutions and changes, 

 , d u u A 2 n -J 1 — e~ ( — 2 e sin d „ tan 2 H tan L sin u cos T > 



12 dx l3 c ll^-ecosd (H—tanH)cos 3 DsinHy 



The last term may be further simplified by multiplying and dividing by sin T, then 

 substituting sin D for sin a sin T, and — cos H for tan L tan D, and so cancelling 

 tan H, as shown in the result which follows. Referring to (20), and collecting the 

 terms of summation represented by X u, we obtain the annexed general expression 

 of the Sun's intensity for any assigned part of the year; thus, 



