66 DIFFERENTIAL AND INTEGRAL CALCULUS. 



is so simple, and this equation is of itself so useful, that 

 we may well insert it. Suppose we measure a distance 

 x (fig. 11) along a fixed straight line Ox, from a fixed 

 point 0, and at the end of any distance x erect a per- 

 pendicular to Ox, of length = ((#), the ends of these 

 straight lines or ordinates will trace out a curve of some 

 form, PQ suppose ; let OM = x, and, therefore, 



then LQ = NQ-NP=$(x + h) - </> (x) , 



Now whatever the form of the curve PQ, provided it is 

 continuous and does not move to an infinite distance 

 between P and Q, the tangent at some point between P 

 and Q must be parallal to PQ, at E suppose, so that if 

 RTbe the tangent at E, LETx = L QPL, but if A 7 be the 

 value of OU the abscissa of R, tsLi\ETx = (f>' (X) as was 

 proved in the first chapter ; and since X lies between x and 

 x -|- li, we may denote it by x + 6h, so that the equation 

 gives us 



i.e. $(x + h) = <t>(x) + h<f>' 



6 being a positive proper fraction, and < (x] a continuous 

 function of x, which does not become infinite between the 

 values x and x -f h. 



This equation alone furnishes proof of the theory of 

 " Proportional Parts" in taking logarithms &c. from tables 

 and indicates the exceptional cases, which are when <' (x) 

 is either very large or very small, so that its changes 

 between x and x + h are either themselves very large ; or 

 are large compared with the whole difference wanted. 



The limiting value of 6 in this equation, when li is 

 indefinitely diminished, is always \. For 



< (x + h) = (x) + hf (tf + 



