( 282 ) 



For n — 1/3 to V2I/3: 



\~2n"- / 1 



[/iV—n- from 7^ — ^ to J/ 1— n- [x from —7, ^ to 1 



2l/l— n'-2 V 4(1 — w^) 



For n = 1/0 1/3 to 1 



\/if~7i'^ from — l/l— 7*2 to l/l— ?r (.<: from 1 to 1). i 



Substitution of these limits in (13) gives, paying attention to the 

 circumstance tliat wliw'cver ^''a: — n^ appears in <a«— ^ , the value of 

 tan—^ is equal to n for l'^* — n^ = — l/] — ifi , when at the same 

 time l/l — X (which becomes 0) appears in the numei'ator: 



7„ (n — Va to 1/0 1/3) = — (Va + % n^ — % "') 1^3—4 n^ + 



1 — 2 n-* ?A2 10 / 1 — an- 



21 1/3— 4 n2 39 V/3— 4?i3| 



; ia»(~' n tarc^ 



n 10 2n 



' (14) 



h {n = 1/3 1/3 to 1) -- ;r 71 A n^ n^' A n'' + 



' \.2n 10^2 2 ^ 5 J ^ 



+ Vo 71 I 3 (2 — 3 n + «S) t/i_^„^ _^ 

 + 2 ;i (2—3 « + n^) \/\—n^ (tan~^ 



l/l— «••2 



Here I have also availed myself of the circumstance that 



—n 1'- 



tan~^ . — 2 tan—'^ 



I/..'— «2 V/^_l/, 



1 , 1_2 n^ 



disappears for x = ^^ ( Vx — m- =: 



4(1 — «2) V 2i/l_n2, 



The expressions found for /„ and Ii have been verified by me in 

 various ways and every time found true. The\ are both still to be 



multi})lic(l by ^'o, n 11'' — _- . Befoi'c passing to the second integration 



with I'ospect to n, we must calculate a compktneiifarij term, for the cases 



