DIFFERENTIAL AND INTEGRAL CALCULUS. 



the curve crosses the axis of x at at an angle of ^TTJ 

 and also does the curve y = tan# (tig. 2). 



Another good illustration of the meaning of the word 

 limit is given by considering what is meant by saying that 

 a railway train, which may be continually varying its speed, 

 is at any given moment moving at the rate of so many 

 miles per hour. Every one, I believe, has a very clear 

 conception that this is so that at any one moment the 

 train is going at one particular speed j but if we try to see 

 how this is to be defined we are led at once to that par- 

 ticular kind of limit which is called a differential coefficient. 



Suppose that during t minutes the train has gone over 



Q 



s yards, then if the rate were uniform - would be the 



t 



number of yards described in one minute, and this is true 

 however large or however small t may be ; whereas, if 



o 



the speed be variable, - will be continually changing, and 



will only represent the average velocity during the portion 

 of time t. In this case if we take t continually smaller 

 and smaller, this fraction will approximate more and more 

 nearly to the velocity which the train has at the middle 

 of the time t ; and the speed at any particular moment 

 will be the limit of this fraction when t is indefinitely 

 diminished. Thus, to fix the ideas, suppose the train is 

 so moving that the space described during any time , 

 measured from a certain epoch, shall vary as the square of 

 , say s = a' 2 , then if during an additional time t' the space 

 described be s', we shall have s + s described in the whole 



time 1 -f ', or s + s = a (t + tj\ hence s = 2att' + a*' 2 , or 



j 



- 2a.t-\- cut' i.e. the average velocity during the time t' 



t 



is 2otf-t ', and diminishing t' indefinitely we obtain the rate 

 at the end of the time t to be 2af. If we now change our 

 notation a little, putting As for s', A for ', we have 



A 9 * 



= 2#2-t-a.A, and if we denote the fact of taking the 



