146 Mr. Herschel on circulating functions, &c. 
At all events however several inconveniences embarrass this 
method. It is entirely at variance with the uniformity which 
ought to reign throughout all analytical operations, thus to 
descend into arithmetical details in the outset of symbolic 
investigation, and to vary our processes according to the 
numerical form of the quantities concerned. If we would 
avoid this by having recourse to an equation of a higher 
order including all the separate cases, we are required either 
to seize at once, by an undirected effort of intellect, on the 
relation which connects the terms periodically equidistant, or 
to go through some preparatory process to discover it, which 
for the most part will be found very troublesome. More- 
over, it demands the actual formation of certain terms to 
determine the constants, which are not (as will hereafter 
appear) necessary. It is true that in the very simple case I 
have just stated, these inconveniences, though really existing, 
are not felt. If we take one a little more complicated, they 
will speedily form a prominent part of the difficulty. Suppose 
the law of formation of the terms of a series were, for instance, 
as follows : 
« u 09 u 3 — bu 2 -f- /3 u l ,u A ~cu i -i r au z , 
U s = au A + (Su v us = bu s 4 ~ oM y u 7 = cu 6 -f- , &c. 
it would be necessary to divide the whole investigation into 
six cases, and to integrate six several equations of differences, 
u 6x + 2 t= ~^6x+ i”j~ a '^6x> ^6x-\- y =: bu6 x ^. z -\‘ll>Usx+ i> &C. 
and after all, the general term of the series would not be 
obtained, but merely the several general terms of six other 
series which, interlaced, as it were, one with the other, form 
the series in question ^ which is in fact much the same way 
