ON CERTAIN GEOMETRICAL OPERATIONS. 67 
whatever be a and 6* positive or negative and with the ordinary 
convention 
Je =) 
It is obvious that 
ne! R2a+1 = R 
[2 = 1 [2at] = ff 
Gizas == «\) Tr+l_~7T 
so that, if we consider any transformation whatever of C, in the 
representation of it the letters 2, Zand 7’ need not appear at any 
power but the first. Moreover by considering the table I have 
previously given, we see that by simple transformations any two 
of the letters #, 7 and 7 being consecutive can be replaced by a 
single letter. Therefore two of the letters 2, J and Z7' need not 
appear consecutively. Consequently any transformation of C 
whatever will present itself under the form : 
5 See ara SS i) Ea i) CRs MR A 
the different powers of P being separated by one of the letters J, 
# or T so that we can consider the expression as being composed 
of different expressions of the form 
Fee ig 1 2Y 
Jeg ea 
LEO TE SZ 
We will now show that any of these expressions can be put under 
the three forms: 
POR Poh Pry 
Let for instance be JOEY Fite JE 
lie oand b-s.-0 as Pk=T we have 
UNG, ties ee 2D 
andas [P= 
JET TS EN ig ols 
then obviously 
JO iD NO ass) let I fad BAe 
and ifa > b 
VERY ON MT el 16) gp thus * =a—b 
* Of course a and 6 are whole numbers. 
y+ Assuming that kis smaller than the smaller of the two numbers « 
and b or equal to it. 
