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CALCULUS 
If v D > v R , then c < 1, and the duck will make port. If, however, 
v D <v R , the duck will be carried down stream and approach the 
opposite bank asymptotically. Finally, if v D = v B , then c = 1, and 
the path is a parabola. The duck will sidle up toward a point half 
as far down the bank as the breadth of the stream — much as 
the Rhine steamers make a landing. 
(c) The Dog and IDs Master. A dog, out in a field, sees his 
master walking along the road and runs toward him. Find the path 
of the dog. It is assumed that the dog always heads straight for 
his master, that each moves at a uniform rate, and that the road is 
straight. 
The same figure can be used as in the case of the tractrix, but 
with a different interpretation. For here, 
OQ = v M t, 
Differentiating with respect to y and observing that ds/dy is nega 
tive, we find: 
The integral of this equation has been found for the case that 
c ^ 1 and y — 1 when x = 0 ; § 5, Ex. 2 : 
and since, for a given value of c, all curves are similar, this one 
gives the shape for the whole class. 
If c < 1, the dog overtakes his master at the point of the road for 
which x == c/(l — c 2 ). If c > 1, the dog approaches the road asymp 
totically. The case c — 1 presents no difficulty; but equation (6) 
is replaced by one in which a logarithmic term appears. 
A number of further problems similar in character to those dis 
cussed here are given in Tait & Steele’s Dynamics of a Particle, 
Chap. I,