OF THE COMPARISON OF VARIABLE QUANTITIES. 



SECT. II. OF THE COMPARISON OF VAKI 

 ABLE gUANIITIES, 



44. Definition. The quantities by which 

 two variable magnitudes are increased or de- 

 creased in the same time, are called tlieir in- 

 crements or decrements, or tlieir increments 

 positive or negative. 



f' Scholium. They are denoted by an accent placed over 

 the variable quantity ; thus x' and y' are the simultaneous 

 increments of x and y. 



45. Definition. The ratio which is the 

 limit of the ratios of the increments of two 

 quantities, as they are taken smaller and 

 smaller, is called the ratio of the velocities of 

 their increase or decrease. 



Scholium. It vcould be difficult to give any other suf- 

 ficient definition of velocity than this. If both the quan- 

 tities vary in the same proportion, the ratio of x' and y' will 

 be constant (16), and may be determined without consi- 

 dering them as evanescent ; but if they vary according to 

 different laws, that ratio must vary, accordingly as the time 

 of comparison is longer or shorter : and since the degree of 

 variation, at any instant of time, does not depend on the 

 change produced at a finite interval before or after that in- 

 stant, it is necessary, for the comparison of this variation, 

 that the increments should be considered as diminished 

 without limit, and their ultimate ratio determined ; and it 

 is indifferent whether these evanescent increments be taken 

 before, or after the given instant, or whether the mean be- 

 tween both results be employed. 



46. Definition. Any finite quantities 

 in the ratio of the velocities of increase or 

 decrease of two or more magnitudes, are tiie 

 fluxions of those magnitudes. 



Scholium. They are denoted by placing a point over 

 the variable quantity, thus, x, y. And.^ is always ulti- 



y 



mately equal to-:-. The variable quantity is called a fluent 



y 



with respect to its fluxion, as x is the fluent of ic, or .rr:./i. 



On the continent the term fluxion is not used, but the 

 evanescent increment is called a difference, and denoted by 

 d or 3, and the variable quantity is conceived to consist of 

 the entire sura or integral of such differences, and marked 



/, as x:r:/dx, otJUx. This mark has the advantage of dif- 

 fering in form from the short s, which is used as a literal 

 character. 



47. Theorem. When the fluxions of 

 two quantities are in a constant ratio, their 

 finite increments are in the same ratio. 



For if it be denied, let the ratios have a finite difference ; 

 then if the time in which the increments are produced be 

 continually divided, the ratio of the parts may approach 

 nearer to the ratio of the fluxions than any assignable dif- 

 ference, for that ratio is their limit (46), and this is true, 

 by the supposition, in each part ; therefore the sums of all 

 the increments will be to each other in a ratio nearer to 

 that of the fluxfcns than the assigned difference (3 5). 



48. Theorem. The fluxion of the pro- 

 duct of two quantities is equal to the sum of 

 the products of the fluxion of each into the 

 other quantity. 



Or (ry)-^yi+xy. Let the quantities increase from 

 X and y to ar-fa' and y-\-y', then their product will 

 be first xy and afterwards .ri/-|-j/.r'-t-j)/'-(-a;y, of which 

 the difference is yi^' -^xy' -^rx'y' and the ratio of the in- 

 crements of x and xi/ is that of x' to yx'+xy'+x'y' ; or, 

 when the increments vanish, to yx'+xy' since in this case 

 x'y' vanishes in comparison with xy'. But x'-(j/.r'-f-jy');;i' 

 (yi+iy), and the fluxion is rightly determined (46) ; 



y' V J^v ^v , , , yx' yx , , 



(orsmce ^ = ^, —, —-^ (18); but:^=4- (is), 



XXX X X X 



and 



yx'-\-xy' yi-^ry 



(IS). 



49. Theorem. The fluxion of any power 

 of a variable quantity is equal to the fluxion 

 of that quantity multiplied by the index of 

 the power, and by the quantity raised to the 

 same power diminished by unity. 



Or (x")'=:7ix""'i. Let n—i, then (xx)-=:x.i-)-xi (48) 

 ::=2xi=nx"~'i. l(n—3, x"— (xx).x, and its fluxion is x 

 (xx)'-i-(xx)i::r2xxi-fxa:i:^3x'i::;iix""'i'. And the same 

 may be proved of any whole number. If 71 is a fraction, 

 1 



as — , put yzzx", then x^yf, and xzzpy^''y,y::z — — - zz 



— . j'"'i(38)z:-y. j-^i— 7U[""'i, as before ; and in the 



same manner the proof may be extended to all postible 



cases. 



