492 PROCEEDINGS OF THE AMERICAN ACADEMY 



now described by the generatrix is called a tangent line to the curve, 

 in accordance with the following general definition : The tangent line 

 to a curve at a given point is the line passing through the point, and 

 having the direction of the curve at that point. 



IV. The Ratio of the Rates is independent of their Absolute Values. 



Since the direction of the curve (or of the tangent line) at the point 



having a given abscissa is determined by the form of the function, or 



By 

 equation, to the curve, the value of — , which is the trigonometrical 



tangent to the inclination of this direction, must be independent of the 

 arbitrary quantity Dx, which merely determines the velocity of the 



generating point. 



D ( f x) . D (f,x) 



In general, the value of — \r' J will change with that of x ; — ' ' 



is, therefore, independent of Dx, but is generally a function of x. 

 D (f x), when expressed in terms of x and of D x, is of the form 



D (f x) = (j> x . Dx 



in which <£ x is another function of x. 



In the ordinary methods, the introduction of an equivalent proposi- 

 tion is, for the most part, avoided by rejecting from the ultimate value 

 of A (/, a:) all terms containing powers of Ax higher than the first. 



We shall now proceed to deduce, from the four elementary proposi- 

 tions hitherto proved, the differentials of the functions both algebraic 

 and transcendental. These propositions are here recapitulated for 

 convenience of reference : — 



I. D{x-\-h) = Dx. 



II. D(x-\-y) = Dx + Dy. 



III. D (mx)=.m D x. 



D( f x) 



IV. — ir ' ■ is independent of D x, but is generally a function of x. 



The Differential of the Square. 



Let z = x-\-h, then Dz = Dx, [1] 



and z 2 = x 2 -\-2hx + h' 2 , and D (z 2 ) = D (a: 2 ) + 2 h D x. [2] 



Dividing [2] by [1], ^ ■ = —l^-\-2h,ands'mceh = z — x, 



