1 74. Dynamics. 



of generating in a second by its action repeated equally during 

 each instant of this second, must be less, and vice versa. 



282. From the two equations ds =v dt, dv = g dt, we can 

 obtain a third that may be employed with advantage. Thus 



from the equation d s = t> d t, we deduce d t = ; substituting 



this value instead of d t in the equation dv = g d t, we have, 



ds 



or, 



g d s = v d v. 



283. We remark, that in the process by which we have just 

 arrived at the equation d v = g d t, we regarded the velocity as 

 increasing. If it had gone on diminishing, it would have been 

 necessary, instead of d v to put d v ; so that the two equations 

 dv = g d t, and g ds = v dv, to become general, must be written 



-h dv = gdty d=igds=zvdv, 



the upper sign being used when the motion is accelerated, and 

 the lower when the motion is retarded. 



284. There is a fourth equation that may be deduced from 

 the two fundamental equations, and which should not be omitted. 



Thus, the equation d s = v d t gives v = -j- ; whence we obtain 



substituting this value for d v in the equation gdt= d i>, we 

 have 



If we suppose, as we are authorized to do, that d t is constant, 

 we shall have, 



gdt = -j^-or gdt* = dds. 



