512 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[537 
Trace the two curves 
a 
a 
V =|(e* + e *), 
shown respectively by the black line and the dotted line in fig. 2. Draw any line 
parallel to the axis of x, meeting the first curve in the points P, P' respectively, 
and let the ordinates MP, M'P' meet the second curve in the points Q, Q' respectively; 
Fig. 2. 
V 
x 
0 M N M' 
then it is clear, that if for a given value of l the line PP 1 can be drawn in such- 
wise that MQ + M'Q' = l, there will be in fact the required two values c = OM and 
And since for MP very large we have MQ, and therefore also MQ + M'Q', very 
large, and as MP diminishes, MQ + M'Q' also diminishes until it attains a certain 
minimum value, say = \, it is clear that if l has any value greater than this minimum 
value, PP' can be so drawn that QM + Q'M' — l. 
[The above remarkably elegant investigation in regard to the two values c, c' 
was given in the Examination; it seems to be the case that as PP' moves downwards, 
MQ + M'Q' continually decreases (viz. MQ decreases more rapidly than M'Q' increases), 
its value being least, and =2NS when PP' becomes a tangent to the first curve at 
its lowest point R; but it is not by any means easy to prove that this is so. The 
question depends on the form of the curve defined by the equations 
a 
a 
a 
a 
X = \x x (e Xi — e Xl ) + \xi (e* 2 — e x,> ), 
a 
a 
a 
a 
Y = 2 x i ( eXa Ye x *) = ^x 2 (e x2 + e x *) 
where X and Y are the current coordinates.]