441 
860] 
DIFFERENTIAL EQUATION F (u, dll/dz) = 0. 
loop ; and, writing y = 0, we have 4# 2 — 27P = 0, that is, x 1 = 0, the node at the origin, 
2 
and x = + g—^, the other two intersections with the axis of x. The tangents at these 
two points are vertical; as being on the axis, they are thus permissive points. And 
they are the only points with a vertical tangent; in fact, the point (x = 0, y= oo) 
is a point of the curve, with the line oo as an osculating tangent (4-pointic inter 
section) ; hence the tangents from the point x = 0, y = oo are the line oo counting 
four times, and the lines x — ± . touching at the points (± 0 — ,0) as above. 
o V 'J \ oyo / 
For the infinite branches, we have y = 3« 4- ..., which is 
1 
y = X 1+Vn P 
of the right form 
We thus see that there is no prohibitive point; and the differential 
equation has a monotropic solution accordingly. 
Writing z in place of z — c, the solution in fact is 
u = 
e z — e~ 
e z + e~ z \e z + e 
e~ — e 
e z — e~ 
hence, putting 0 = z — —, the curve is given by the two equations 
x = 0-0\ y = (l — S0 2 ) (1 — 6 2 ), = 1 — 4# 2 + 30 4 : 
the monotropic function u satisfies the differential equation. 
( d U \ 
u, = 0 must be either a rational function, a singly periodic function, 
or a doubly periodic function of z\ say the forms are 
u= P (z), u = P {e gz ), 
and 
= P {sn (gz, &)] + cn (gz, k) dn (gz, k) Q {sn (gz, A?)}, 
where P, Q denote rational functions. I do not at present consider the criteria 
(such as are given in Briot and Bouquet’s Theorem V., p. 301) for determining by 
means of the curve which is the form of the integral. I remark, however, that in 
the first and second cases the curve is unicursal, while in the third case it is 
bicursal; or say that, according as the deficiency is = 0 or 1, the integral is rational 
or simply periodic, or else it is doubly periodic. Moreover, the curve being unicursal, 
we can express the coordinates as equal to rational functions P (6), Q (6) of a parameter 
6; and, being bicursal, we can express them as functions of the form 
P (0) + Pj (0) Vl — 0-. 1 — Q (0) + Q, (0) Vl - 0 Z . 1 - 
of the parameter 0; and supposing the coordinates (x, y), that is, u and T , thus 
du 
dz’ 
expressed, it should be easy to establish the relation of 0 with z, e 9Z or sn (gz, k) 
in the three cases respectively. 
C. XII. 
56