346 A. Mukhopadhjay — Memoir on Plane Analytic Geometrij. [No 3, 



Hence, from (183), we get 



a _ /* rdr 



P' J |2a(n-l)+ /[rH4a8(l-^)]r' 

 If, therefore, we take p for all values of r from+ oo to — oo , we have 



Jo, [ ^"^ ^'' ~ ^^ "^ /[^-H4a3 (1 - n)] l ' ^^^^^ 

 To evaluate this definite integral, let us first take the indefinite form. 

 Put 



r2 = 4a2 (1-%) tan2(/), (185) 



r=z2a\/l — ntB.n<l>, 

 dr = 2a \/l — n sec^ <t> d<p, 

 r8-|-4a2 (1 - 7i) = 4a2 (1 - ti) sec* (f>. 

 If, therefore, I be the indefinite integral, we have 



/* 4a2 (1 — 7^) tan <l> sec* </> c?(^ 



J [2a (71-1) + - "" 





2a x/l — oi sec <^ 



4a* (1 — n) sin </> d<p 



2a \/l — -^i — 2a (l — n) cos </> [ 

 4a* (1 — w) sin </> # 

 8a3 (l-w)^|l->v/r3i;cos</>| 



1 /* cZ (cos </>) 



2a^/^^J ji-^j— ^cos<^ 



1 1 



4a (w — 1) 



j 1 — \/l — n cos <^ I 



(186) 



Now, sec* = 1+ tan* </> = !+ ,^ from (185) . 



4a* (l-7iy ^ ^ 



Therefore 



cos*0=^i^L(lz^ 

 r* + 4a* (l-7^)' 

 and, when r = na, this gives 



„. 4(l-n) 



(2 — w)* 

 and, when r = oo , 



cos * </) = 0. 

 These give the limits of the transformed integral; if, therefore, Q be the 



