Astronomical and Nautical Collections. 323 



the particular purpose to which they are immediately applied, and 

 it is not likely that the uses here assigned will ever be so generally 

 adopted without express definitions, as to be worth imprinting on 

 the memory. 



4. " The characteristic £ implies the sum of a finite number of 

 terms, derived from all the possible variations of a quantity, which 

 is here denoted by a small letter of the Greek alphabet." 



Laplace, and others, have used the t in this manner ; and it is 

 easy to observe the distinction of expressing the quantity, which 

 is variable to a certain extent by a small Greek letter, thus 

 pA+ Bx + Cx*... dx _ ^y y ^ a '_• . where u tQ be made 

 J a+ bx + ex* .. . f 



successively equal to each of the roots of the equation = a + bx 

 + ex 9 . . . , v = A + Bg + C ? 8 . . . , and f = b + 2ct> + 3rf ? 3 . . . 



5. " A comma , in an index, denotes or." 



A comma is also used by some writers to signify and, in ex- 

 pressions of what they call functions of different quantities. 



Again, Cel. Mech. p. 76. " In this country it has been usual, 

 at least till very lately, to preserve the geometrical accuracy intro- 

 duced by the great inventor of the method of fluxions, and to call 

 any finite quantities, in the ratio of the velocities of increase and de- 

 crease of two or more magnitudes, the fluxions of these magnitudes. 

 Thus, if we call the increments of x and y, x' and y\ we have, for the 

 fluxions, any magnitudes x and y, so assumed, that x : y shall be 

 equal to x' : y' when these increments become evanescent. On the 

 Continent, it has been more common to write dx and dy for x' and y', 

 considered as actually evanescent ; and Euler has remarked, at the 

 beginning of his Integral Calculus, that the language of the Eng- 

 lish is the more correct, but that the continental notation is the 

 more convenient. His words are these : — l Quas enim nos quan- 

 titates variabiles vocamus, eas Angli, nomine magis idoneo, quan- 

 titates fluentes vocant, et earum incrementa infinite parva seu 

 evanescentia fluxiones nominant, ita ut fluxiones ipsis idem sint, 

 quod nobis dirTerentialia. H»c diversitas loquendi ita jam usu in- 

 valuit, ut conciliatio vix unquam sit expectanda : equidem Anglo- 

 iu formulis loquendi lubenter imitarer, sed signa, quibus nos situ 



