346 A Theory of Fluxions. 



consequence. In conformity with this idea, they have called 

 it an infinitesimal, to express in a strong manner the small- 

 ness of the quantity from which they might be allowed to ex- 

 punge the part MEK. But however small it may be, it has 

 still some magnitude, and so far it must, at least in theory, 

 be considered as involving an error. 



Sec. 6. Fluxions of the higher Orders. 



When, in the generation of a variable quantity, its fluxion 

 is different at different stages of its production, it may be 

 considered as a fluent, and its fluxion taken, which is called 

 the second fluxion. Also when the second fluxion is a vari- 

 able quantity, the fluxion of this fluxion may be taken, which 

 is called the third fluxion. After the same manner, fourth, 

 fifth, sixth, &-c, fluxions will arise, when the preceding ones 

 are variable quantities. The first fluxion of x" is 2xx', its 

 second fluxion, considering the fluxional base x' a constant 

 quantity, is 2ar 2 . The first fluxion of x 3 is 3x 2 x', its second 

 fluxion is 6xx' 2 , and its third fluxion is 6a;* 3 . In each of these 

 instances, the last term becomes a constant quantity, and 

 the next fluxion is equal to 0. It hence appears, that the 

 index of the power of a variable quantity points out the high- 

 est order of fluxions, which that power admits of. The re- 

 lation of the fluxions of the several orders to the correspond- 

 ing increment of a variable quantity, may be discovered in 

 the following manner. Suppose ar to be the increment of 

 x, then '* 



Ixx'-^-x' 2 is the increment of a; 2 , 



and 2a;a;-+2x'2+0 are the orders of the fluxions, 

 again Sx 2 x'-\-3xx' z -{-x' 3 is the increment of x 3 , 



and 3a; 2 x , -f-6xx' 2 -f-6x #3 +0are the orders of the fluxions. 



By comparing the terms, which compose the increment, 

 with the corresponding terms, which constitute the first, se- 

 cond, third, &c, fluxions, it will be found, that the powers 

 of x and x' are the same in each ; therefore if the" several 

 terms in the latter quantity be divided by certain divisors A, 

 B, C, &c, it will become equal to the former, and the cor- 

 responding terms themselves will be equal ; that is, — — -f 



A. 



J!l_=23Er- r -a: ,a . Taking the corresponding terms, — ■ *= 

 B A 



