438 
Proceedings of Royal Society of Edinburgh, [july 18 , 
these forms. The groups into which the integers from 0 to 3 + s 
must be divided are 
| 0 | 1 2 | 3 4 | 3+* 
and for the case of %a a bPcyd s 
the duads are 
0, y + 8 + 1 ; 1 , y + 8 ; ; 8, y + 1 : 
the triads 
0, 1, /3 + y + 8 + 2; 0, 2, /3 + y + 8 + 1 : 
and therefore the number of dovetailings 3, which is the required 
coefficient. 
For the case of %a a b&cy the only duad is 
0, y + 1 ; 
the triads are 
0, 1) 0 + y + 2 ) 0, 2,/3 + y+ l, 
and consequently the number of dovetailings is 2, as was required 
to be shown. 
For the case of %a a b&, there is again only one duad 
0 , 1 ; 
and two triads 
0 , 1 , 0 + 2 ; 0 , 2 , 0 + 1 : 
and therefore manifestly only 1 dovetailing. 
Our first theorem is thus established. 
The second is — The expansion of \ a% 2 c 3 d 3+s | + 1 a°b l c 2 d 3 | or 
a(0, 2, 3, 3 + s) consists of the single symmetric function 2; a*bc and 
all the like functions succeeding it, the coefficient of every term 
involving three letters being 1, and the coefficient of every term 
involving four letters being 3 (ii.) 
For example, 
a ( 0 2 3 6) = %a%c + %a 2 b 2 c + 3 %a 2 bcd . 
Here only two forms of terms require to be considered, viz. 
%a a b$cy and %a a b$cydP. The groups into which the integers 
0, 1, 2, . . . . , 3 + s must be separated are 
| 0 1 | 2 | 3 4 5 | 3 + *. 
For the case of %a a b$cyd 8 , the duads are 
0, y + 8 + 1 ; 1 , y + 8 ; ; 8, y + 1 : 
