INTENSITY OF SUN'S HEAT AND LIGHT. 25 



It may be remarked that the arc H can be developed in terms of its cosine ; A 2 

 may be expressed in powers of cos 6 ; and thus u may be represented entirely in terms 

 of the true longitude T; and ultimately in terms of the mean longitude or anomaly; 

 as, u = A + B sin (b + ax) + C sin (c + 2 a x) + D sin (d -f 3 a x) + . . . ; where a or 

 n denotes the Sun's daily motion in longitude,~"or arc 59'8" ; which is .0172. This 

 arc being so much less than unity, shows that the regular process of summation 

 without a second constant, will converge with extreme rapidity, stopping at the 

 first differential co-efficient, and leaves us at liberty to determine the sum 



/u dx + \ u + T V — in such manner as may be most convenient. 

 dx 



Therefore, let x or t denote the number of days elapsed after the beginning of the 

 year or epoch; n being the mean daily motion in longitude ; a' + n t or a' + n x, the 

 mean anomaly; T, the true longitude, and P the longitude of the perihelion, so 

 that the true anomaly = T — P, and d 6 = d T. Also, if a denote the obliquity 

 of the ecliptic, then by Astronomy, sin D = sin o sin T. 



Since cos H = — tan L tan D, we have sin 2 H = 1 — tan 2 L tan 2 D, or again 

 cos 2 L cos 2 D sin 2 H = cos 2 L cos 2 D — sin 2 L sin 2 D. Substituting in the last mem- 

 ber, 1 — sin 2 D for cos 2 D, also 1 for cos 2 L + sin 2 L; then dividing by cos 2 L, and 

 taking the square root, 



cosDsinH= h_£^D |l_(^)%^r. (21.) 



N cos 2 L \ \cosL? K ' 



With respect to A 2 , let us here write its values from equation (8), and another 



value given by the ordinary polar equation of the ellipse ; assuming A to be 1 ; and 



c, a new constant such that, since d 6 is equal to d T, 



A 2 = 1 = cdT - = c^ + ecosdf 99 , 



p 2 ndxV\ — e 2 (1 — ej ' K ' J 



Substituting now the third members of the last two equations in place of the 

 first members which occur in the preceding expression for u, and multiplying by dx, 



udx= \ sin L sin u sin T.H + cos L f 1 — (. " ) sin 2 T t . (23.) 



n<Sl—e 2 l S \cosLJ S J 



The next step is to integrate this equation, where in the first term, sin a sin T 



has been substituted for its equal, sin D. The integral of the last term is readily 



sin o 

 identified as the arc of an elhpse whose eccentricity is j ; therefore let 



COS ±j 



j d T 11 — ( — y) sin 2 T = E, an elliptic function of the second species. 



Again, integrating the variable factors of the first term by parts, Csin T . Hd T = 



— H cos T + C cos T d H. To obtain d H in a function of T, let us differentiate 



sin D = sin a sin T, and cos H = — tan L tan D, giving cos D d D = sin a cos Td T, 



tanLdD tan L sin u cos T d T 



and sin Ha H = 2 7-, . Whence d H = —. — ^ ^ 2 -^=r ; or substituting 



cos 1) sin H cos D . cos- JJ & 



