241 
of Edinhurgh, Session 1882-83. 
Conversely, the snms of the series (4) and (5) are known ; for 
they are respectively that is 
and 
4. To obtain results corresponding to (3) in the case of a cubic 
and of equations of a higher degree, we observe that in the case of 
a quadratic the number of terms in {p + qY is the same as the 
number of different li's represented by • 
Taking the equation 
= px^ + + r 
we have 
K = P ’K-i-^qK-2-^r . . ( 6 ), 
where hn denotes the sum of the homogeneous products of roots 
of the cubic. By the repeated application of (6) we deduce 
K . K-i + 2i?2' . K-% + . . . + . 7?,,_s 
~{p + q-\-rf . symbolically, 
where we observe that occurs twice. 
Again, it will follow that 
'»»=(?+«+»•)'• 
the number of terms in {p-\-q-\- r)^ being ten, while there are only 
seven different /i’s, /i„_ 5 , hn-a, /^«_7 each occurring twice. 
So it will follow that 
[h]zi 
= (i, + 2 + r)‘.[/7]::'. 
and generally 
ha = {p-\-q + rj . [/?,] 
n - 
> 
the number of different /i’s being 2r + 1 , and so disposed that in 
the first pair each h occurs once, in the second pair each Ji occurs 
