856 
MAJOR P. A. MACMAHOX OR THE THEORY 
{02 20)J 
(n n) 
(01 21) j 
(I2l0)j 
(H ^ oT)l 
(01 01 20 ) 
(10 02 10 ) 
(IT To ol) J 
1, r = 2 
5=1, r = 3 
(01 01 10 10)1 
(ro“ oT To m) 
(^ To To ^) 
(To ^ M To) 
y S= 1, r = i 
(oT To TT) 
(^ To To) 
(TT ^ To) 
(To ^ TT)J 
^ 6- = 1, r = 1 
(01 11 To) 
5 = 2, r = 3 
(01 10 01 10 ) 
r = 4. 
The compositions present themselves in pairs of conjugate groups according to the 
several values of s and r. In the above example for 5 = 1, 7’= 3, there is a self¬ 
conjugate group. This happens when 2?" = p -|- </ + s + 1. 
A self-conjugate group exists for even or uneven values of s, according as -j- q is 
uneven or even. 
21. The enquiry now is in regard to the number of lines of route through the 
reticulation which possess exactly s essential nodes. 
It may be observed, in passing, that from any line of route may be derived a 
composition whose graph exhibits only essential nodes and no others. This may be 
called the principal composition along the line of route; it will have 5+1 parts, and 
each of the s contacts of its jiarts will be positive-positive. 
There is a one-to-one correspondence between the lines of route having s essential 
nodes and the compositions of 5 + 1 parts, all of whose contacts are positive-positive. 
Also the number of compositions, number of parts unrestricted, all of whose 
contacts are jiositive-positive is equal to the number of different hnes of route. 
E.g., for the number 22, 
s = 1 
s — 0 
Lines of route 
3 
Compositions (22) (02 20) (ll 11) (01 21) (12 10) (01 11 10) 
and there are no other compositions having all contacts positive-positive. 
22. The number of different lines of route with exactly s essential nodes is 
