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Proceedings of the Boycd Society 
twice, in tlie third three times, and so on up to the ^th pair, while 
T 
the middle /?, i.e., hn-^r^ occurs n + 1 times. The same law holding 
when we reckon from the end, the total number of terms will be 
4^1 4-2 + 3 + . .. + 2^ + 2 + 1, 
that is, ^ 1) 2), which is the number of terms in 
{p + q-^ ry. 
Again, if we consider the equation 
■x^ =px^ + qx^ + rx + , 
we have 
hn=P • K-l + 2' • ^ • ^^n-3 + 5 . ^n-4 ) 
from which it may be deduced in the same manner as beforh that 
+ ^q . /?n- s + • • • + 
= (p + g + r + s/ . [hjfl symbolically, 
in which , /r„_ 5 , /i„_ occur twice. 
So it will follow that 
hn = {p-\-q + r^sf . 
{p + q-\- r + sf . [h]l 
and generally 
^n = (p + q + r + sY .[h]f I, 
the number of different 7z’s being 3r + 1 , which are repeated by the 
same law in going from the beginning and the end, and such that 
the sum of the number of all their coefficients is equal to the sum 
of the series, 
l+3 + G + 10 + ... + ^(r+l)(r + 2) 
that is 
(r+l)rr + 2) (r + 3) 
1.2.3 
which is the number of terms in (jj + ^ + r + sy . 
