in Mathematical Reasoning. 367 
s — tv’ 
v+v- 
If A begin to move after B, the time ¢ must be made negative, 
and then these two cases become 
s — tv’ 
= i= ? 
and 
s — tv’ 
merorwe’ 
the former of these referring to the case of the bodies moving in 
the same direction, and the latter to that of their direction being 
opposite. 
There are some restrictions which ought to be noticed, if 
the velocities of the two bodies are equal, or v = » the first and 
third cases show that the bodies can never meet. To this there 
: : a ; ) 
is however an exception, if v =», and also s =/v’, then = 
an indefinite expression and « may have any value: the signi- 
fication of this is that both bodies move in the same direction 
and with equal velocities, since v' =», and that the hindermost 
B, which starts t seconds before the other, is situated at such 
a distance s from it, that it arrives at the point where the other is, 
exactly as it begins to move; this appears from the equation s=¢v’, 
it is therefore obvious that the two bodies will be at the same point 
at every part of their progress and for every value of x. Whenever 
x is negative they can never arrive at the same point in the di- 
rection in which they move. If however we conceive that they 
had been moving at the same rate prior to the point of time 
at which we consider them, the negative value assigned to « marks 
a point through which they both passed at the same moment. 
These two modes of translating the same question into 
Algebra, and of re-translating the result into ordinary language, 
