Harmonics of the Second Type. 17 



Now, if r is odd, 



rv/2 . 77- 



j cos 2r(j) (log cos <£ — log sin <£) dcf> = ~- 



by a well-known formula, and we find 



(^(cos 6) cos 20d0 = £ + £ = | w , 

 Jo 2 b o 



or finally, 



2.4. ...4n + 2 



j; 



Q2»+i (cos (9) cos (2 w + 2) dd = it . 



3.5. ...4n-r3 



_,_ 2 4» + a (^ + l!)' 

 (4n+3!) 



(41) 



By applying the reduction formula for the trigonometric 

 function, we also have 



I r = ] Q2n +1 (cQS0) cos {2n + 2 + 2r) 6 dO 



_ 1.3. ... 2r-l 4/z + 4.4?2+6 . ... 4?2 + 2r + 2 T 



~~ 2.4....2r *~ 4n + 5. ... 4n + 2r + 3 0> 



so that 



f *" Qa»+i (cos (9) cos {2n + 2 + 2r)6 



d<9 



= 2r± (2n + r+ll)» 24jl+2 f42> 



(r!) 2 ' (4n + 2r + 3)! * ' v ; 



Accordingly, between zero and it, Q2„+i admits the Fourier 

 series 



Q 2n+1 (cos(9) 



= 2 to + 3 V oV ^ cos(2n + 2)0+£.p^cos(2n + 40) 



4n + 3 ! \_ y 2 4n + 5 v y 



, 1.3 4n + 4.4n + 6 , .. ^ ,._ 



which ceases to converge at the limits. 

 We now consider the integral 



j Q 2B (cos0)cos(2?i+l)0d0. 

 Phil. Mag. S. 6. Vol. 43. No. 253. Jan, 1922. C 



