361 
of Edinburgh, Session 1876-77. 
The corresponding limit in the case of other functions may be 
similarly interpreted; that is to say, the limit of if such be 
possible, when n is indefinitely increased, is a root of the equation 
<£(#) = #. We may view such limits as implying an infinitude of 
operations ; and we have seen that when this infinitude of operations 
is carried out upon any value (within certain limits) of the inde- 
pendent variable, the result is always the same; in other words, 
we have seen that those functions, in the case of which the limit is 
possible, are levelling functions, having the property of bringing 
everything they act on to the same dead level. In this respect 
they may be compared with Euler’s expression 
1 . 1 . 0 1 
2 # + sin x + 2 sin Ax + ^ sin 3# + .... 
7T 
which for all values of x within certain limits equals ^ ; and this 
suggests the possibility of finding a similar expression for cos 00 # 
by means of Lagrange’s or Fourier’s theorem. 
7. Note on Determinant Expressions for the Sam of a 
Harmonical Progression. By Thomas Muir, M.A. 
(Received February 27 — Reau. March 1877.) 
Taking the harmonical progression 
a a + b a + 2b a + 3b 
and denoting the sum of n terms of it by S n we have by means of 
Euler’s transformation, 
S 
n — 
1 
-g— + 6) 2 
2a + b ~ ' 2 ^+ 36 . > + 26) 2 
2 a + 5b — 
{a + {n - 2 )6} 2 
2a -f (2n — 3)6 
a - 
