866 Mr Greenhill, Integrals expressed hy [Mar. 8, 



the integral 



= 2 



d<^ 



V(r' + 2rs cos 2 </) + §') 



2 [ dcf) 72 _ '^'^^ 



r + sj V(l-/i='sm» ' (r + sY 



2 , (x — m + ntano) 2 J(rs)] 



= arg tn ■{ -. — ■- — , — ^^-^ — > . 



r + s [n — [x — 7n) ta.n (JL> r + s } 



For instance, as particular cases of (22), 



dx , /I — x"^ '. 



= A arg' en 



/, 



dx 1 f^/S — X 



Thus 



oV(^' + 8a?-''+3) 2^3 





J^/(a 



i^{a + hx + CO? + dc^ + ea?*) 



which can be reduced to one of the forms (10) to (22), can be 

 expressed by inverse elliptic functions. 



By the inverse notation the results of the integration of all 

 integrals, a number of which are considered in Legendre's Fonc- 

 tions Elliptiques, which by substitution can be reduced to elliptic 

 integrals, can be written down ; thus as examples 



- arg en {-—^ — ^^ o -, - , sm 15 \ , 



VCl-^') 2;y3 ^ 1(V3- 1)^^+1 



[ (1 - xY^ dx (2-^ > a; > 0) = f' (l-^^)"^ dx{l>x> T^) 



U^ {{^J^ + \):j^x'{l-x'f-\ . -.J 



= jT^-^ arg en \— — ' ^ ^ -, — - , sm lo" 



24/3 ° 1(V3-1)3^^(1-^«)U1' J 



(Bertrand, Calcid Integral, p. 686). 



[J (1 + xY"- dx = V2 arg en |(1 + x'f^ , -^1 , 



j^ (1 - xY' dx = V2 arg en |(1 - x')'^ , ^| , 



1% ^ ^^-l^ ri-7^^3i 1] 

 I (x^ — 1) ^ dx = arg en ■{ , , -T^r , 



