Daily Variation of Solar Radiation, etc. 539 



Remembering that 



fi 1 \n+s 



(15) implies 



j\ ,f±? ^P« (-l) t(w - 8) .2g [„ + * |i Q>-a)|l(p + g) 



V h > * d^» T 2".1.3...(n+p+l).2A..O>-n)i(n-«)ft(n+*) 



(17), 



with w, jj, 5 odd or even together, ?i = s, p = s. p —n; but if with 

 n, /?, 6' odd or even together, s > p, then the integral vanishes. 



In the same way (16) gives for n and s odd or even together 

 and p the reverse, and either n or p the greater, 



7r.1 .3... (// +yj) w + g (s - p)(g - p + 2) . .. (g +p) 



2 n+1 . 2 . 4>...(n+p+l) \ H n ~ s ) \ j(n + s)( n -p)(n -p + 2)...(n + p) 



(18)- 



For the ® notation the connecting formula is 



[» - *(1 -/*»)'«P n "(/Q= 1 . 3. ..(2-/t- 1)(h) h ,,. 



Two particular cases may be noted, viz. those for which 6 = 0; 

 (16) then gives for p = - 1, 0, and all positive integers* 



(p = — 1, opening term - j 



§** 7T r.3...(2p + i) , 7T . rao + ni' 



(l-M 2 ) =2 • 2T47:(2^r2) + 2 ( - 1) ll-»-(^P + 1 )] 



»=• (4w+l)1.3...(2n- 1)1.3... (2w+2p+l) p 



• h 2 . 47. .2w.2.4...(2w+2p +2)(2w-2p -1 )(2n -2p+l). ..(2w+2p +1 ) "^ } 



(19), 



and similarly (15) gives 



fi- tt M(i«ys- (-i)»(4n+i)P m o*) /: (20) 



* This expansion was obtained directly in Messenger of Mathematics^Stet. 1896, 



p. 92, but the reduction there given is only true for 2n>2p + l. The form above 

 suits either case. \ 



