104 



A. Mukhopadhyay — On Poissons Integral. 



[No. 1, 



Now consider the general integral 



r"^ cos 2nx dx 

 ~ Jo 1 + cos'^ X ' 

 which gives 



du _ r'^ X sin ^iix dx 



dn Jq l-\- cos^ X 



Hence 



Again, 



(du\ _ r''^ X sin x dx rr^ 



dn/^^i Jo l+cos^o;"" 2 



= 1 — cos^ x-\- cos* X — &G. 



l-fCOS^ £C 



Therefore, the (y + 1)*^ term of u may be written in the form 



( — 1)^ I cos 2??^ cos^** X dxy 

 Jo 



which at once leads to the symbolic relation 



§. 4. Geometric Interpretation. 



It is interesting to remark that the analytical transformation in 

 Poisson's remarkable relation (5) easily admits of an elegant geometrical 

 interpretation from well-known properties of the ellipse. 



Consider the semi-ellipse 

 APA', of which S is a focus, 

 and C the centre ; AQA'is the 

 semi-circle described on AA' 

 as diameter ; take any point 

 P on the ellipse, join PS, 

 draw PM at right angles to 

 AA', meeting the circle at Q, a 

 and join QC. Let the angles 

 ASP and ACQ be represented by 6 and u, respectively; then, as usual 

 in the theory of elliptic motion, we have the famous relation between 

 the true and eccentric anomalies, viz., 



cos u — e 



cos 6 =: 



1 — e cos u 



whence 



sm 



e=\/i- 



1 - e cos u 



(U) 

 (12) 



