322 Astronomical and Nautical Collections. 



the subsequent letters generally for quantities considered as variable. 

 They are here employed as relating indifferently to quantities po- 

 sitive or negative, and to numbers, whole or fractional, except 

 when they are used as indices or exponents." 



It may be remarked that the use of the letter o is generally 

 avoided, to avoid confusion from its similarity to 0, or zero : though 

 printers, even the most careful, are continually in the habit of 

 printing an italic o instead of zero. 



2. " The Italic characters are employed in preference to others, 

 for denoting quantities in general ; the Roman for characteristic 

 marks, as d for a fluxion, or differential, sin, cos, or f, 9, for sine 

 and cosine ; and hi for hyperbolic logarithm. The long italic J*, 

 however, not being otherwise used, serves very conveniently as a 

 characteristic, to denote a fluent." 



The index of a power is regularly marked by a smaller character 

 above and to the right of the root, as o? for aa, x 3 for xxx ; and in 

 the same manner it is usual to write d«, instead of repeating the 

 characteristic d, because of the frequent occurrence of a second 

 fluxion: but it has been more common to write cos a ai for (cos x) 2 * 

 rather than for cos cos x, since the latter expression can scarcely 

 ever occur, though Mr. Herschel thinks the symbol would be more 

 correctly employed in this sense. In the same manner it is usual 

 to write f*<S?x for the second fluent of the second fluxion of x, and 

 it seems to be more distinct, and therefore more elegant, to write 

 3 y<K ydx for the particular fluent of ydx, taken between the values 

 x = 2 and x = 00, than to place the 2 on the right of the y*, ./**<» , 

 as Mr. Fourier and others have done. 



3. " When the Italic letters m, n, p, $ r, or any others, are here 

 employed as indices, they are to be understood as denoting any 

 numbers without limitation ; the Roman small letters, m, n, will be 

 applied to whole numbers only, excluding fractions, but either po- 

 sitive or negative, or 0; the small Italic capitals M , y, to positive 

 numbers, whether whole or fractional, excluding negative numbers 

 only ; and the small Roman capitals, m, n, to positive integers only, 

 including, however, 0." 



Distinctions of this kind are only intended to be recollected for 



