CHAP. H.] SUPPLEMENT TO CHAP. VII. 107 



/x=2h 

 tt.vi 



n th< 



r 



x = h, I a 

 Jh 



differential of y to differential of * a , or 5-, is the " second differential co- 



d x 



and when * = 2 h 



and by subtracting one from the other, we have for the integral between 



* 7 



the limits * = 2 A and x = h, I tf d x = h*, and O thus disappears. 



'h 



We have, then, only to substitute in succession the values of the variable 

 which indicate the limits, and subtract the results, 



If also there is but one limit, we could determine O if there were also a 



condition, such, for instance, as that / a; 2 d x should equal Ti when x = 2 A. 



The ratio - is called the " first differential coefficient ; " if it were to 



d x 



be differentiated again, the next ratio, viz., that of the differential of the 

 differential of y to di: 

 efficient," and so on. 



Thus, y = x s ; d y = d (x s ) = 5 x*d x, or -= = 5 ** ; differentiating again, 



ct x 



= 20 x 3 d x, or ^ = 20 x 3 , and so oh to third differential coefficient, etc. 

 ax dx* 



7. Example. As an example of the application of our principles, let 

 it be required to determine the area of a triangle. Let the base be & and 

 the height h. Take the base as an axis, and at a distance of x above the 

 base draw a line parallel to b, and at a very small distance d x above this 

 line draw another, thus cutting out a very small strip. (Let the reader draw 

 the Fig.) Now for the base y of this strip we have the proportion Ji x : y 



::Ti:l t or y = I , hence the area of the strip is & dx . But 



n h 



the area of this rectangular slip is not equal to the area of that portion of 

 it comprised within the triangle. It projects over at each end, and is, 

 therefore, somewhat greater. Thus for the small trapezoid actually within 

 the triangle we have for the upper side y', h(x+d x") : y':: h : b, or y' = 



b- (x + d x). Hence y y' = -, and the area of the projecting portion 



fi hf 



of the rectangle, that is, its excess over the trapezoid, is then (yy") d x, or 



bdx* _, , ,, bxdx b d x* , do, , b x bdx . 



. Therefore, o d x ^ = d a, or - = = , where 



ft h h, d x n h 



d a is the area of the small trapezoid itself. Now these latter two quanti- 

 ties are always equal for any value of d x. But as d x decreases, one side 



of the equation approaches the limit & r-, and - , therefore, approaches 



n ax 



this same limit. The rectangle itself is, then, the limit of the ratio of the 

 area of the small trapezoid to its height, and we can then equate the limitt 

 themselves, remembering that in this case d a is the area passed over by the 



