24 INTENSITY OF SUN'S HEAT AND LIGHT. 



Here u — ar°; Cu dx = t . 



J log r 



du xl d " U .1 3 C 



j — = a r log r ; -j— ; = a r x log r ; &c. 

 doc b dx° & 



The sum of the coefficients of a f being constant, let it be denoted by B ; then 



will 



2 « = Ba r % + C. 

 If x = 0, a r — B a r + C. 

 If a? = 1, = _B a + C. 



ft n* # *T" * — //y» 



Whence 2«= ^ ; which also agrees with the well known rule. 



r — I ° 



III. To find the sum of the trigonometric series, 



sin a + sin 2 a + sm 3 a + .... + sm a? «. 



Here w = sin x a; Ca d x = cos x a. 



J a. 



du d s u 



-j — = a cos x a ; -j— » = — a cos x a ; 

 dx ' dx 



proceeding, therefore, as in II., we have 



2«= i sin xa 4- B cos xa + C. 



If £=(), = 5+C. 



2 w = £ swa a? a + B (cos xa — 1). 



If x = 1, sm a.= £ » a + B (cos a — 1), 



,, , sin a . cos i a 



And B = i 



cos a — 1 sin J a 



~ i . cos i a (cos x a — 1) 



Sm = 5 sin xa — -^ — ■ '- 



2 sin i a 



Reducing to a common denominator, we have by Trigonometry, 



cos ha — cos (x + i) a sin (x+ 1) i a sin \ x a 



2 sin i a sin i a 



The formula of summation has its failing cases ; but these may be pointed out 

 as plainly as those of Taylor's Theorem. Without entering here into a full dis- 

 cussion, it must apply in all cases where the summation is in its nature possible, 

 and the differential co-efficients do not become infinite. It applies rigorously where 

 the terms are all positive, and the differential co-efficient becomes zero, as in 

 Example I. ; also where the collective co-efficient can be represented by a second 

 constant, denoted by B, and so can be eliminated, as in Example II. and III. Had 

 not advantage been taken of this feature in the last Example, the sum were repre- 

 sented by the following series, which still converges rapidly when a does not much 

 exceed unity : 



2 u, or 2 sin x a = — — cos xa+ hsin xa + -^a cos xa — T ^ 7 a 3 cos xa + .... + C. 



Having now demonstrated the formula of summation, let it be applied to (13) 

 where the diurnal intensity is measured by 



u = A 2 (sin L sin D.H + cos L cos D sin H). 



