of Edinburgh , Session 1876 - 77 . 
245 
IV. A great many other deductions from the fundamental pro- 
position are given — for instance, 
A closed plane curve, intersecting itself, divides the plane into 
separate areas whose number is greater by 2 than the number of 
intersections. 
Regarding the curve as a wall, dividing the plane into a number 
of fields, if we walk along the wall and drop a coin into each field 
as we reach it, each field will get as many coins as it has corners, 
but those fields only will have a coin in each corner whose sides are 
all described in the same direction round. The number of coins 
is four times the number of intersections — and two coins are in 
each corner bounded by sides by each of which you enter— none 
in these bounded by sides by each of which you leave. 
Cut off at any intersection and remove a portion of the curve 
forming a closed (not self-cutting) circuit. You thus abolish an odd 
number of intersections. Hence if there is an even number of 
coils, whether the whole be clear or not, there is an odd number of 
intersections, and vice versd. 
To form the symmetrical clear coil of two turns and of any (odd) 
number of intersections, make the wire into a helix, and bring one 
end through the axis in the same direction as the helix (not in 
the opposite direction, as in Ampere’s Solenoids ), then join the ends. 
[The solenoidal arrangement, regarded from any point of view, has 
only nugatory intersections.] An excellent mode of forming this 
coil is to twist a long strip of paper through an odd number of 
half-turns, and then paste its ends together,- — the two longer edges 
become parts of one continuous curve which is the clear coil in 
question. This result is applied to the study of the form of soap- 
films obtained by Plateau’s process on clear coils of wire. 
Y. Another question treated is the numbers of possible arrange- 
ments of given numbers of intersections in which the cyclical order 
of the letters in the 2nd, 4th, 6th, &c. places of the scheme shall be 
the same as that in the 1st, 3rd, 5th, &c., i.e, the alphabetical. 
Instances of such have already been given above. In the first of 
I (5), for example, the letters in the even places are 
D E AB C. 
Here the cyclical order of the alphabet is maintained, but A is 
postponed by two places. 
