94 Scientific Proceedings, Royal Dublin Society. 
Or, since 
sin 6= sin A sin Z, 
where 
A= obliquity of ecliptic, 
? =sun’s longitude, 
= 2A Tw : ° 
= sind sind - sin/ 
ee m4 
+ cosh {18 9 (1 = cos 21) =" A(3 - 4 cos 27 +008 42) ~ 
6 
a (10 — 15 cos 27+ 6 cos 4—cos 61) — ke. 
bas a 
+ tan) sin A = = (i — cos 20) + Sd — 4 cos2/ + cos4/) + a1) 
ino 
z aly —4 cos 21+ 6 cos 4/—cos 62) + &e. 
sintA 
192 
( sin’A 
(2560 
: 3 sinSA : £ 
+tan?r sin) 4 (3—4 cos 2/ + cos4l) + 9 (10-&e.)+c&e. 
5 
+tan°A sind 
(10 —&e.) + &e. 
+ &e. 
We must now substitute for 7 the sun’s longitude, on each day 
of the year, and add the 365 terms together; this will convert all 
the periodic terms of (11) into the sums of sines and cosines of 
ares in arithmetical progression taken all round the circumference, 
and with a very small common difference.* 
The periodic terms, therefore, vanish in the summation, and we 
obtain, 
The heat received in one year = 2A sin XE sin 6(H —tan Z) 
=2A x 365-25 sin2A 68sintA 5sinsA 
-|cosA< 1 — = aS = a, 
4 64 256 
(sin?A 3sintA 15 sin'A 
+ &e. 
+ tan dr a 7 + aot Re Bl 
na | a 
+ tan®r sin BP PIAS ee (12) 
64 256 
SA 
tan®A sin X Seg Xe. 
+ tan°A sin 256 + We | 
+ &e., &e. 
Substituting for A its value, 23° 28’, we obtain, finally, 
* Equal to 59’ 8” (the daily change in sun’s longitude), or small multiples of that are, 
