CHAP. n.~J SUPPLEMENT TO CHAP. VH. 109 



For x = 0, C = 0, and the area above the axis is then b h. For as = and 



18 



SB = A, we have for the area below the axis b A. This area has a dif- 



3 18 



ferent sign because below. If we give it the same sign as the other, and 

 then add it, we have the total area. If it also had been above, the total 

 area would have been the difference. Generally, then, we subtract accord- 

 ing to our rule. 



. Significance of the first differential coefficient. Any 

 equation between two variables of the first degree is the equation of a 

 straight line. If of the second degree, it represents one of the conic sec- 

 tions, an ellipse, circle, parabola, or hyperbola. Of a higher degree, a 

 curve generally. If, then, we take the axis of x horizontal and y vertical, 

 and if d y and d x are the consecutive increments of y and x, that is, the dif- 

 ference between any value and the very next, the ratio - is evidently the 



cL x 



tangent of the angle which a tangent to the curve at any point makes with the 

 horizontal. 



If, then, we make = 0, and find the value of the variable * corre- 

 ct x 



spending to this condition, we find evidently the value of x for which the 

 tangent to the curve is horizontal. If now the curve is concave towards the 

 axis, this value of x, substituted in the original equation, will give the maxi- 

 mum or greatest value of the ordinate y ; because for the point just one 

 side of this the tangent slopes one way, and for the point just the other 

 side it slopes the other. The point where the tangent is horizontal must 

 then be the highest. 

 If the curve is, on the other hand, convex to the axis, the value of x, which 



makes -=- = 0, substituted in the original equation, will give y a minimum 

 a x 



value for similar reasons. By setting the first differential coefficient, then, 

 equal to zero, we may find that value of x which corresponds to the maxi- 

 mum or minimum value of the ordinate, as the case may be. In the case 

 of the deflection of simple beams upon two supports, the curve is always 

 concave to the axis, and hence we obtain by this process always the maxi- 

 mum deflection. 



The above comprises all the principles of which we shall make use in the 

 discussion of the theory of flexure. With -a little study, we believe that 

 any one familiar with analytical operations, even although he may never 

 have studied the differential or integral calculus, can follow us intelligently 

 in what follows. Whatever points may still be a little obscure will clear 

 up as he sees more plainly than now their application. 



