58 A Discourse on the Theory of Fluxions. 



left the theory in some parts incomplete. The humbler at'- 

 tempt is all that I aspire at, of clearing away the rubbish, 

 and rendering easy and pleasant to the learner, the entrance 

 to this science, by exhibiting a view of its first principles. 

 That this may be fully accomplished, I shall repeat a part of 

 what was laid down in the before mentioned communication, 

 but somewhat varied in the mode of illustration, and con- 

 taining a more full statement of this part of the theory. 

 By a series of fluxions, or fluents, is to be understood that in 

 which each term is a multiple, composed of an invariable 

 factor, and one that is variable consisting of one or more 

 terms, with, or without invariable coefficients. Here the in- 

 variable factors may be of any assignable magnitudes, pro- 

 vided they differ from each other ; but the variable factors 

 must be the same, or of equal value. When the invariable 

 factor is not expressed, it is considered as being unity. It 

 will be found on examination, that a fluxion is equal to a mul- 

 tiple, composed of its corresponding fluent, and the quantity 



71X' 



— . From the use which is made of the factor x- in this 



X 



quantity, it is obvious, that it may be of any finite magni- 

 tude, great or small, that can be assigned. To make use of 

 this equation in explaining the relation under consideration, 

 let Bx" and ax" be two functions of the variable quantity :r. 

 Their corresponding fluxions nBx"~^x' and nax''~^x' may 

 be considered as two terms, selected from a series of fluxions, 

 constructed in conformity with the foregoing definition ; 

 which selection may always be so made, that one of the 

 teriiis shall be the fluxional expression, that occurs in the 

 process. The factors B, and a represent the invariable part. 

 The magnitude of the variable part is determined by the 

 limits assigned to the fluents at the time of their production. 



Theorem. I. Any two terms, in a series of fluxions, will 

 have their corresponding fluents in the same ratio with them- 

 selves. 



Theorem. II. Any two terms, in a series of fluents, will 

 have their corresponding fluxions in the same ratio with 

 those fluents. 



Hence, 7iBx^~^x- : B:r": :nax" ^x- : ax^. 



Let the antecedent be represented by A, the consequent 

 by C, and the ratio by r. Then by the definition of a ratio 



pi =r. Any two of these being given the other may be ob- 



