332 
Proceedings of the Boyal Society 
Its symbol is always of the form 
2 r 3 + (n ~ 3 )r 2 
2 + 
| or more fully 
/i\ 
0*-l) = (w-l), 
there being n — 2 lines in the lower group. 
The three last forms have each essentially only one degree of be- 
knottedness. In certain cases (see the foot note ante p. 296) we may 
give two degrees of beknottedness by altering some of the signs 
— but the knot has then one nugatory intersection, and falls into 
the class with five crossings. 
A number of curious problems are suggested by the process 
which I employed in the investigation of these six-crossing forms. 
I give the following as an instance. 
Take any arrangement whatever of the first n letters : — Say, for 
instance, 
CND A . . . LE. 
change each to the next in (cyclical order, so that A becomes B, 
B becomes C, . . . . , N becomes A) and bring the last of the 
row to the beginning. The result is 
FDAEB . . . . M. 
After performing this operation n times we obviously get back the 
arrangement from which we started. [Thus in seeking all the 
different forms of knots of a given number of crossings, one alone of 
this set of n need be kept.] The problem is to find sets such that 
the original combination is repeated after m operations like that 
above. It is obvious that if m is to be less than n it must be an 
aliquot part of it, and thus n must be a composite number. 
\_Ayril 11. — The references to Listing’s type-symbol here given 
must be taken in connection with the extracts from his letter, ante , 
p. 316.] 
3. Laboratory Notes. By Professor Tait. 
(a.) Measurement of the Potential, required to produce Sparks of 
various lengths, in Air at different pressures, by a Holtz 
machine. By Messrs Macfarlane and Paton. 
The general result of these strictly preliminary experiments 
appears to show that for sparks not exceeding a decimetre in length 
