122 
MAJOR P. A. MACMAHON OX A CERTAIX CLASS OF 
vA'hile its first coaxial minors have each the value 
(_)«-! _ 2), 
and its coaxial minors each the value 
1). 
Hence the generating function is 
{1 
_ V 
~ — ... — — ... — (?t— 1) x-^x^ . . . x^} 
or writing 
this is 
( /y* ■ I I O'* O'* I ('y* O^ I — /■/ r/>‘^ 1 I /--/ -v*'^ “ ^ 
1 — «2 — 2a.^ — oa^ — ... — (?i, — 1) cin 
Art. 17. As another example, again consider the permutations of the quantities in 
O* O* ^2 O* 
I • 
Divide the places occupied by the quantities into compartments 
Al Ag . . . A;,, 
such that the first places are in compartment A^ 
next 4 ,, „ Ao 
last 
and let us find the number of the permutations which liave the property that no 
quantity with an uneven suffix is in a compartment with an uneven suffix, and no 
quantity with an even suffix is in a compartment with an even suffix. 
In the “ particular redundant generating function ” we have merely to jiiit 
rq = rq = a- = . . . = 0 , 
= bg = . . . = 0, 
Cl = Cq = Cq = . . . = 0, 
&c., &c., 
and the remaining a, b, c, . . . letters equal to unity. 
