583 
of Edinburgh , Session 1874 - 75 . 
6-2862. Wherefore, if each difference have been computed true 
to the nearest last figure, the maximum error arising from this 
mode of calculation is 31431. M. Lefort, taking into account the 
maximum possible error in the first logarithm, makes it 3-95, or 
say four units in the last place. 
All this looks exceedingly well, but has not the slightest reference 
to the matter in hand. In order to obtain such a miserable degree 
of precision, we have the labour of computing the first difference 
of each order, and then the toil of writing 6 times 12 times 200, 
or 14 400 unnecessary figures ; for, to make M. Lefort’s formula 
applicable, each difference of each order must be carried to the 
26th place. 
Prony did not use the method of differences; he used a method 
of vitiated differences. To show the nature of the vitiation, I 
transcribe a few lines of the actual work from M. Lefort’s example, 
which, belonging to an advanced part, has only differences of the 
fourth order. 
Nombres. 
Logarithmes. 
A 1 
A2 
A 3 
A* 
100 800 
00346 05321 0951 
43084 5563-17 
4274 19-79 
848-03 
2-52 
801 
00346 48405 6514 
43084 1288-97 
4274 11-31 
848-00 
2-52 
802 
00346 91489 7803 
43083 7014-86 
4274 02-83 
847-97 
2-52 
803 
00347 34573 4818 
43083 2740-83 
4273 94-35 
847-94 
2-52 
Here we see that the differences, though computed true to the 
last figure, are only used to the second preceding figure; thus 2*52 
is read 3, and the possible error is augmented one hundred times. 
But this is not all; the difference of any particular order only 
comes to affect those of lower orders by the accident of some of the 
to be rejected figures being more or less than 50; so that the final 
effect cannot be made the subject of calculation. I find nowhere 
any attempt to estimate the effect of this systematic vitiation, 
and shall endeavour to supply the want by taking two extreme 
imaginary cases. In the first case I shall assume each of the 
initial differences to be 0-49. Proceeding with these according to 
the method of Prony, we find 
