224 PROFESSOR A. G. GREENHILL ON THE 



According to HALPHEN (' Math. Ann.,' 15, p. 359 ; ' Comptes Rendus,' March, 1879) 



H(nr)_H^^ 



H(2 ( ') J(1 "- 1) 



5. The integral employed l>y ABEL, changed to the circular form, can he written 



Z = z - (Az* + Kz + C)- (2), 



and, as pointed out in the ' Archiv der Mathematik und Physik,' 3. Reihe, I., p. 72, 

 this is exactly in the form required in the problem of LEVY'S ' Elastica,' with z = r-/a~. 

 We reduce it to our standard form in (4), 1, hy putting 



. _"(") *(3*>) _. , // /ox 



' ,s (,,)-,x(r) ' s + x 



Z = ,-(A^ + B 2 + Cr- = -- (4), 



so that 



(A*-' + B* 



_ - --- 

 x -f ./ (s + ,r) 4 



- x)+ x^- 



4 (., + i j* 



X 



A-.- + B: + C = 1 - + (6), 



2 (x + x) ^ -2 (x + xf 



and therefore 



A - - r R - 2r + .'/ + //" ( ' - x y + v~ t7\ 



i!r ' -2>r t!f 



A + B + C = I (8), 



and : 1 is a factor of Z corresponding to * = ao . 

 More general ly, with 



so that 1/M is the value of z corresponding to x = oo ; then in ABEL'S notation, 

 changing the sign of z and Z to obtain the circular form, 



(10), 



_ -- 8 JL 



M x - * ( wv ) 4 [s - x (nt') }* " 



