FLUXIONS. 



or a 



1 n 



; that of a X ~s* 



= ; and that of 



m X n + 



In assigning the fluents of given flux- 

 ions, it ought to be considered whether 

 the flowing quantity, found as above, re- 

 quires the addition or subtraction of some 

 constant quantity, to render it complete : 

 thus, for instance, the fluent of n xni 

 may be either represented by .m or by x 

 a ; for a being a constant quantity, 

 the fluxion of xn a, as well as of x", 

 is .r n _U\ 



Hence it appears that the variable part 

 of a fluent only can be assigned by the 

 common method, the constant part being 

 only assignable from the particular nature 

 of the problem. Now to do this the best 

 way is, to consider how much the variable 

 part of the fluent, first found, differs from 

 the truth, when the quantity which the 

 whole fluent ought to express is equal to 

 nothing; then that difference, added to, 

 or subtracted from, the said variable 

 part, as occasion requires, will give the 

 fluent truly corrected. To make this 

 plainer by an example or two, let y = 

 3 x. Here we first find 



but when y = 0, 



becomes = ; since x, by hypothesis, 



is then == : therefore always 



exceeds y by ; and so the fluent, pro- 



a-j-:^ a 4 

 perly corrected, will be y = 



, 3 a 1 x 1 , , x+ 



= a3*H + a*3_|-_. Again, 



let y = a^-hrm," x ^""ix : here we first 

 have y = -r-^and making y = 0, 



the latter part of the equation becomes 

 e^n+l j.mn+m 



w ,Xttl = m s/n-4-i ' w ^ence the equa- 

 tion or fluent, properly corrected, is y = 

 a,m -j-a-m^ n+1 a mn+m 

 mXn+l 



Hitherto x and y are both supposed 

 equal to nothing at the same time ^ 

 which will not always be the case : thus, 

 for instance, though the sine and tangent 

 of an arch are both equal to nothing, 

 when the arch itself is so ; yet the secant 

 is then equal to the radius. It will there- 

 fore be proper to add some examples, 

 wherein the value of y is equal to nothing, 

 when that oi'x is equal to any given quan- 

 tity a. Thus, let the equation y = x 1 x, 

 be proposed; whereof the fluent first 



found is y 



- 

 o 



but when y = 0, then 



~, by the hypothesis; therefore 





the fluent, corrected, is y = 



a-3 a? 



Again, suppose y = xn x ; then will y 



^.fi+i 

 = ; which, corrected, becomes v *= 



- . And lastly, if y = c3 -f~^)i 



X x X-, then, first, y = c*+b x^ : therC- 



3 b 

 fore the fluent, corrected, is y = 



3. To find the fluents of such fiuxionary 

 expressions as involve two OP more vari- 

 able quantities, substitute, instead of such 

 fluxion, its respective flowing quantity ; 

 and, adding all the terms together, di- 

 vide the sum by the number of terms, and 

 the quotient will be the fluent. Thus, 



the fluent of x- y-f y 3;== ^ ^ ~ ^ 



-O Ai 



= xy; and the fluent of x yz+yx z+z y x 

 x y z + xyz -\- xy z^Sxy z ^ 



3 3 ' X y 



But it seldom happens that these kinds 

 of fluxions, which involve two variable 

 quantities in one term, and yet admit of 

 known and perfect fluents, are to be met 

 with in practice. 



Having thus shown the manner of find- 

 ing such fluents as can be truly exhibited 

 in algebraic terms, it remains now to say 

 something with regard to those other 

 forms of expressions involving one varia- 

 ble quantity only ; which yet are so af- 

 fected by compound divisors and radical 

 quantities, that their fluents cannot be 

 accurately determined by any method 

 whatsoever. The only method witk re- 

 gard to these, of which there are innume- 



