212 Dynamics. 



if we imagine the velocity MO, which gravity tends to give in 

 an instant, decomposed into two parts, one MD perpendicular to 

 MN, and the other DO or ME directed according to MN ; we 

 shall see that it is by virtue of this last that the velocity of the 

 body will be accelerated. Now by letting fall the perpendicular 

 RN, and comparing the similar triangles MOE, MNR, we shall 

 have, 



M JV : JVfi : : M : ME = ^/ O . 



Let us suppose that the different points of the curve JIB are 

 referred to the vertical axis BZ. If we call, 



BP, x ; PM, y ; and the arc BM, s ; 

 we shall have 



PQ or RN = dx ; and MN = d s. 



We give the sign to these quantities, because x and s go on 

 diminishing, while the time t increases. Let g be the velocity 

 which gravity gives to a free body in a second ; g d t will be that 

 267. which it would give in the instant d t. We shall therefore have 

 the velocity represented by MO as follows, namely, MO = gdt. 



Calling -o the velocity which the body has when it arrives at 

 M-, dv will denote the augmentation received during the time 

 dt-, thus, 



ME = dv. 



Substituting the values above obtained in the equation, 



_ AMX.MQ 



MJV ' 



we shall have, 



But, by article 280, ds = v dt, or d t = - , or, s being consid- 

 ered as decreasing while t increases, d t = - ; whence, by 

 substitution, 



