

ISOPERIMETRICAL PROBLEMS. 581 



that might be prescribed ; the minuter the 

 accuracy to be attained the greater the labour, 

 of course. You must not imagine that I suggest, 

 as a thing of practical engineering, the attain- 

 ment of minute accuracy in the solution of a 

 problem thus arbitrarily proposed ; but it is 

 interesting to know that there is no limit to 

 the accuracy to which this ideal problem may 

 be worked out by the methods which are 

 actually used every day by engineers in their 

 calculations and drawings. 



The modern method of the " calculus of varia- 

 tions," brought into the perfect and beautiful 

 analytical form in which we now have it by 

 Lagrange, gives for this particular problem a 

 theorem which would be very valuable to the 

 draughtsman if he were required to produce an 

 exceedingly accurate drawing of the required 

 curve. The curvature of the curve at any point 

 is convex towards the side on which the price per 

 unit length of line is less, and is numerically equal 

 to the rate per mile perpendicular to the line at 

 which the Neperian logarithm of the price per unit 



