a 1^1 /dn u + k 1 -k\ 171 /1 + 
i = i F lo «(dS^t>'iTfc'j’ s = i tel °g(rr 
1 + sn U\ 
sn u) 
Restoring the radius a, and writing the system in the form 
x= acos 0, y = a sin 0, z= ka.u, 
n , k 2 , /dn u + k' 1 — k'\ , 7 , /1 -f sn m\ 
S: -* U log • iTfc'j ’ s = log (j=W > 
we see that, as u passes from u = 0 to u = K, and therefore 2 from 2 = 0 to 
2 = kaK (k the complete function F 1 » then 0 and s each pass from 0 to oo; 
and, similarly, as u passes from u — 0 to u — — K, that is, as 2 passes from 0 to 
— kaK, then 0 passes from 0 to oo, and s from s = 0 to s = — oo; viz. the curve 
makes in each direction an infinity of revolutions about the cylinder. Developing 
the cylinder, a0 becomes an «-coordinate; viz. we have thus the plane curve 
2 = kan, 
, № a, /dn u + k' 1 —k'\ 
x ~ i Y ]og [m=¥‘T+VJ- 
which is a curve extending from the origin in the direction x positive, to touch at 
infinity the two parallel asymptotes z = ± kaK; and conversely, when such a plane 
curve is wound about the cylinder, there will be in each direction an infinity of 
revolutions round the cylinder.