16 ProL J. W. Nicholson on Zonal 



The ratio of the integrals on the left is already known r 

 and therefore 



(m 4- n) u n + (n — m — 1) u n -2 _ (^ — wi — 1) (n + ni) 

 (n — in) u n +(n+ m — 1) u „_ 2 {n + m — 1) (n — in) ' 



whence we find on redaction 



u n _ (n + m—l)(n — m — 1) 

 ^i-2 (w + w)(w— w) 



Applying this formula of reduction, we can show that 



jQan+i (cos0) cos (2n+2)0dd 



Jo 



2 . 4 . 6 . . . . 2n 2n 4-4.2?i + 6. ..An + 2 



"1.3.5. ./. 2w— l"2n + 5.2n + 7. ...4n + 3 



x y Qi (cos 0) cos (2a + 2) 0d0. 



This integral can be reduced by the preceding reduction 

 formula, in which the trigonometric function is depressed. 

 We find 



f 7r Q 1 (cos0)cos(2n + 2)0^0 



1.3. ... 2n-l 4.6. ...2*i + 2 (V 

 = 2.4. ...2n - ' 5,7/...2n + 3 .1 Ql (C ° S ° ] C ° S *° **> 



antl, finally, the simple form 



] Q211+1 (cos 0) cos (2?i + 2) Odd 

 Jo 



4 6 4;?4-2f^ 

 = :t-f^ Q 1 (cos«)cos2fl^. . (40) 



0.7. ,. 4?i + 6j 



Now 



* Qi (cos 0) cos 2(9 i0 







= P"-^ cos cos 20 log cot 2 | -cos 20~l dO 

 Jo L J * J 



= | (cos 20 + cos 60) (log cos <£ — Iogsin0)d0. (0 = 20), 



