6

5

to study these complicating operations, and to ascertain
precisely what are the cases in which they are indispensible
in proving a graph to be impossible, and to
restrict their employment to these cases. This I have done.
The operations of insertion and iteration receive
their due recognition in Operation 5, while the useful effect of the operation
of putting a double cut is fully attained in Operation 4.
Namely, in case one has [a] [a b], one would
insert a double cut round the right hand a thus,
[a] [ a b ] and then, by deiteration, one would
get [a] [ [ ] b ]. Arrived at this stage
we put a double cut about b giving
[a] [ [ ] b ] and by erasure [a] [ [ ] [ ] ] by
deiteration [a] [ [ ] ] and removing the double
cut [a]. But Operations 4 and 1 accomplish
this and thus obviate the necessity of any