102 SUPPLEMENT TO CHAP. VH . [CHAP. H. 



CHAPTER II. 



PWNCIPLES OF THB CALCULUS NEEDED IN OUR DISCUSSION. 



4. Differentiation and Integration. We need but a very few 

 simple ideas and conclusions in order to have at our disposal the whole 

 theory of flexure for beams of single span. Those to whom these ideas are 

 not familiar already may find them indeed new, but will not find them 

 difficult or even abstruse, and with attention to the following will, we 

 venture to think, make a valuable acquisition. 



The sign / is called the " sign of integration," and integration means 



simply summation. It arises merely from the lengthening of the original 

 letter S, first used by Leibnitz for the purpose. The letter d is called the 

 "sign of differentiation-;" in combination with a letter, as d x, it reads 

 " differential of *," and signifies simply the increment which has been 

 given to the variable x. So much for terms. 

 Now suppose we have the equation 



y = 5*, ......... (i) 



in which x and y, although varying in value, must always vary in such a 

 way that the above equation holds always true. This being the case, let 

 us give to y an increment that is, supposing it to have some definite value 

 for which, of course, x is also definite in value, increase this value by d y. 



Then x will be increased by its corresponding amount d x, and as the 

 above relation must always hold true, we have 



y+dy = 5 (x+dx)* ....... (2) 



or y +dy = 5 (x*+2 x d x + d x*). 



Inserting in this the value of y from (1), we have 



dy = 5 (2xdx + dx*), ...... (3) 



which is the value of the increment of y or d y, in terms of x and the in- 

 crement of * or d x. That is, the increments are not connected by the same 

 law as the variables. The variable y is always 5 times the square of the 

 variable x, but the increment of y is greater than 5 times the square of the 

 increment of x by an amount indicated by 5 x 2 x d x. From (3) we 

 have 



= 5 (2 * + <?*), ........ (4) 



which gives the value of the ratio of the two increments. Now, if we 

 assume a certain value for *, we find easily from (1) the corresponding 

 value of y. If we increase this value of a; by a certain assumed increment, 

 x, we find easily from (3) the corresponding increment of y, or d y. Thei) 

 (4) would give us the ratio of these two increments. 



