300 Proceedings of Royal Society of Edinburgh, [june 20 , 
is a warning to ns against too hasty conclusions from experimental 
data. Let us suppose for a moment that the periodic oscillations 
of a chain had been of great importance ; while as yet we had 
only observation to guide us, means had been found for counting 
the numbers per minute, and it had been found that, within all the 
attainable accuracy these are as the alternate odd numbers, 3, 7, 11, 
15, 19, — this with the greater precision the farther we prosecute our 
trials. Here then we have discovered a periodic law ! Nature has 
a partiality for small numbers, — vibrating bodies divide inharmonic 
ratio, substances combine in easy proportions. So, in seeking for 
the law of light’s refraction, the constancy of the ratio of the sines 
was accepted as absolute, while, as yet, the inaccuracy of the observa- 
tions was so great as to cover the whole range of dispersion. Nay, 
while we firmly hold to the simple laws of pressure and motion as 
revealed to us by our experiments, may it not be that these only 
exhibit close coincidences, resulting from more deeply-seated prin- 
ciples? We know of the law of velocity only from experiment, but 
the velocities observed by us are only relative variations, mere zerre's 
in comparison with the speed with which we and our apparatus are 
hurried along. 
The function which has been under review, and its derivatives 
and primitives, are all deducible from the generic property 
- a.g>z = n. lz g>z + 0 . tefe 
in which, however, n is restricted to be an integer positive number. 
It may be interesting to inquire into the phases when this restriction 
is removed. 
On making the same supposition as before, namely, 
<£z = A + Bz + C z 2 + D^ 3 + , &c., 
we find 
- a A = 1 . 
-aB = 2.(n + 1)C 
- aC = 3 . (n + 2)D, &c., 
whence 
f a Z a 2 Z 2 a 3 Z 3 « 
( f )Z==A \ 1 _ ^T + ^ + l)L2“^+l)(w + 2)l7273 + ’ 
which formula is applicable to all cases except those in which n is a 
negative integer, for then the denominator of a n would come to be zero. 
