A Theory of Fluxions. 333 



inward ; and may be termed the jluxional coefficient. 3. 

 That, in a complete fiuxional expression, there are as many 

 terms as there are variable quantities, that is, in yx there are 

 two, yx'-\-xy', in xyz three, xyz'-\-xzy-\-yzx\ In the square 

 two, xx--\-xx'\ in the cube three, x 2 x^ -\-x 3 x' -\-x 2 x-=3x 2 x- ; 

 and in the biquadrate four, x s x'-\-x 5 X'+x 2 x--[-x s x , =4x :i x', 

 and so on. Hence may be derived rules for assigning the 

 fluxions to fluents, which is called the direct method of flux- 

 ions. But cases exist, in which some of the terms are want- 

 ing, among which is the fluxion of the area in the triangle 

 and curvilineal figures, the formula of which is yx' ; and the 

 fluxion of the solid content in the cone and pyramid, which, 

 although they have three dimensions, have but one term 

 ax 2 x' ; the fluxional base being the fluxion of the height, and 

 the fluxional coefficient ax 2 , the generating superficies. To 

 quantities of this kind, fluxions may be assigned, by consid- 

 ering the manner in which they increase, and setting down 

 their fluxions, instead of their increments. 



In assigning the magnitude of fluxions, mathematicians 

 are not restricted to any particular limits. But although the 

 fluxional base x' is an indeterminate quantity, and may have 

 any assignable magnitude, great, or small ; yet when it is 

 one fixed, all the fluxional bases^ which belong to the same 

 expression, are determined by it : for all the fluxions, that 

 enter into the same equation, must be understood to be pro- 

 duced contemporaneously. 



Sec. 2. Some principles relating to Fluxions. 



Prin. I. If a quantity, varying by insensible degrees ac- 

 cording to the laws of continuity, pass from one state of mag- 

 nitude to another ; it must successively have all the intermedi- 

 ate degrees of magnitude from the least to the greatest. 



Prin. II. No portion of a curve can be assigned so small, 

 but there may be one still smaller. This follows from the 

 infinite divisibility of quantities. 



Prin. III. No portion of a curve can be taken so small as 

 to become a straight line. 



For from Prin. II, no portion of a curve can be taken so 

 small, but one can be taken still smaller, and therefore the 

 portion taken can be divided into two parts ; and if divided 

 into two parts, from the nature of a curve their directions 



