12 INTRODUCTION. 



49. Theorem. The fluxion of any power of a varia- 

 ble quantity is equal to the fluxion of that quantity multi- 

 plied by the index of the power, and by the quantity raised 

 to the same power diminished by unity. 



Or {xn)':=ztLv"-^x. Let 7izz2, then {xx)'=ixx-\-xx (48)=:2a;i=: 

 nx^i—ix. If 7i=i3, x'^zz{xx).x, and its fluxion is x {xx)' -\-{xx)x:=z2xxx 

 ■^xxxzz3x^xi:inxn-^x'K And the same may he proved of any whole 



number. If n is a fracti. -n, as ■ — , put y=.x^y then xzzyP, and xz=. 



V 



X \ 1 



pj/P—]y,yz=: . = — . y^—Px{38):=: — i/. y—'Px:=inxn—\xy as before; 



and in tlie same manner the proof may be extended to all possible 

 cases. 



50. Theorem. When the logarithm of a quantity 

 varies equably, the quantity varies proportionally. 



Or if I a:zzw, ^zz~. For x'Zihy (42), and when y becomes y-\- 

 a X 



y,x+x'=.b^'^-^=by.by'zzx.by', BXidx'zzx.by'—x^ix. f 62''— 1 J; butj; be- 

 ing constant, by the supposition, bv' — 1 is constant, and maybe called 



^, and ar'zzl^ ; therefore x=:-^, and — :zr^. 

 a a a x a 



Scholium. Numerical logarithms do not, strictly speaking, vary 

 by evanescent increments ; but other quantities may flow continually, 

 and be always proportional to logarithms : in either case the propo- 

 sition is true. In Briggs's logarithms, commonly used, b is 10, and a, 

 the modulus, is .4342944819 ; dividing all the system by «, or multi- 

 plying by 2.302585093, we have Napier's original hyperbolical loga- 

 rithms, where ^ becomes =— , and rt=l. 

 x 



51. Theorem. The fluxion of any power of a quan- 

 tity, of which the exponent is variable, is equal to the 

 fluxion of the same power considered as constant, toge- 

 ther with the fluxion of the exponent multiplied by the 

 power and by the hyperbolical logarithm of the quantity. 



Ifxy^^-, z=:ya:y— JA-f-(hl x). xyy; for hi zzzy. (hi a), (42); now 



