I NT 



3. If a body descend down a system of inclined planes, the velocity ac- 

 quired, oa the supposition that no motion is lost in passing from one plane 

 to another, is equal to that which would be acquired in falling through 

 the perpendicular height of the system. 



4. If a body fall from a state of rest down a curve surface which is per- 

 fectly smooth, the velocity acquired is equal to that which would be ac- 

 quired in falling through the same perpendicular height. 



5. The times of descent down similar systems of inclined planes, simi- 

 larly situated, are as the square roots of their lengths, on the supposition 

 that no velocity is lost in passing from one plane to another. 



INFLEXION, point of in curves. 



To ascertain the point of contrary flexure in any curve, find the 2d 

 differential of the equation of the curve, supposing dx constant, and we 



d% y 

 shall have a finite value of ~j~^~ which must be put equal to either 



zero or infinity. By means of this equation, and that of the curve, we 

 can determine those values of x and y, which belong to the point or points 

 of contrary flexure. 



Ex. 1. Let the equation be y = 3 x 4. 18 x% 2 xt. 



t. Let the curve be the cubical parabola, whose equation is yt = at x. 



d?v 2 __1 S. 



Here ^^ = x 3 a 3 = o, .*. x o, or the point of in- 



flexion is at the vertex. 

 For the point of inflexion in spirals see Spirals. 



In general there cannot be a point of contrary flexure, unless the first 

 differential coefficient, which does not vanish, for a particular value of 

 the abscissa, be of an odd order. See Maxima and Minima. 



INTEGRAL. See Differential. 



INTEREST. 



Interest simple. 



Let P principal, r interest of 1. for one year, I the interest of 

 P, and M its amount in the time n ; then we have the following equa- 

 tions, from which any of the quantities may be found, the rst biug 

 jpiven. 



148 



