14 
v-plane AE plane of gia 
sn(v) . dn(v) 
Fig. 5. 
J? —18 = 866, 
T = 6y. 
The equation of the curve transformed in this manner runs as 
follows : 
2 
PALM Si Ws 
+ 
the curve is therefore a rectangular hyperbola. In the cases II and 
IV the §-axis is the real axis, in case VI the y-axis is the real axis. 
Each point of the conic /(v, y) = 0 corresponds to one point of this 
rectangular hyperbola whilst to one point of ®=O two points of 
F=0 are conjugated. The points for which /==0 have as absciss 
S= —}. The line § = — } does not intersect the curve ® in case 
II, but it does in the cases IV and VI. The point at infinity on 
§ +40 represents the points S, and S,; the point at infinity on 
— — n—=0 represents the two points R, and R,. The points 7’, and 
T, are represented by the points of intersection of ®=0 with 
—E—= — } The images of the points 7, and 7, are in case VI united 
in the point of intersection of §= — 4 with the branch of ® — 0 
lying under the S-axis. The images of 7, and 7, are always points 
where the motion changes its sign along the curve ®. 
Now we have to investigate the cases of degeneration. 
Case IIIe. \= 41, d,=0, a,, and a,, not disappearing at the 
same time. 
The point O lies on one of the asymptotes, without coinciding 
with the centre. So this position occurs with the hyperbola only. 
Here equation (71) bolds, in which is put 1, = 0, 
6 
sin T 
died (71) 
Equation (62) (4tb comm. p. 1015) passes, on account of the relation 
