206 Mr Glaisher, On the developments of [Nov. 24, 



Changing Legendre's notation as in the last section, the formula 

 in the Memoires* becomes 



E'=l 



4 

 3 1 



4' 2 k V12 2.3 

 3.5 1.3 F /13_ 



4. 6 "2. 4 \20 2.3.4 4.5.6 

 3.5.7 1.3.5 /19 1 



4.6.8*2.4.6 128 2.3.4 4.5.6 6.7.8 



-&c. 



In this series the general term is 



3. 5. ..(2w-l) 1.3... (2n-3) 



4.6...2/1 '2.4... (2ro-2) 

 where 



6n - 5 1 1 





#.= 



4(2/i-l) 2.3.4 4.5.6'" (2w-2)(2n-l) 2n* 



To identify this term with the corresponding term in the series 

 for E' in § 10, we notice that 



Qn-5 _ 2 _1_ 1 



2(2w-l) 2 2/1-1' 



and —, — — =— -7 — — ?r = : ^ + 



r (r + 1) (r + 2) r r + 1 r + 2 ' 

 Thus 2£„ = 



1 1 /1_2 1\_/1_2. 1\ / 1 2 1\ 



2 2n-l \2 3 + 4/ U 5 + 6/'" \2n-2 2n-l 2n) 



2^3'" 2/i-2 2n-\ 2n" 

 and the coefficient of & 2r ' is evidently the same as in I 2 § (9). 



* Memoire sur les Integrations par arcs d'ellipse, p. 630. 



