MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION. ETC. 47 



wliose sides pass tlirougli the vertices of a given triangle ABC and 

 wliicli shall have a maximum area. (&). To construct a point in a plane 

 at which the sides of a given triangle ABC subtend given angles. This 

 is the familiar coast survey problem of the engineer for mapping harbor 

 soundings. Also it is the problem of orientation by resection of the 

 plane table in terrestrial surveying. The engineer solves this problem 

 without difQculty both graphically and trigonometrically. The well-known 

 constructions for the plane table by Netto and by Bessel can be found in 

 the U. S. Coast Survey papers on the Plane Table. The ordinary trigo- 

 nometric solution of the engineer adapted to logarithmic computation is 

 very simple and as follows : Let A B C he the known triangle whose sides 

 a, b subtend known angles a, p respectively at a point P in the plane. 

 Let x^ LF AC and y ^ LF B C be angles to be determined. In any 

 ease either x-\-y or x — y is known. Then 



PA = I ^-^^(f + ^) , PB=a ^^y±^ 

 sm p sin o 



sin X sin y 



sm /3 sm a 



a sm a 

 b sin /? 



= tan Ii-, (say), 



is known, and by an easy deduction 



tan -1(2; — y)= tan-J(.T + y). tan(fc — 45°), 



determines x and y and solves the problem, (c). Three concurrent forces 

 (vectors) are of constant magnitudes, their lines of action pass respectively 

 through three fixed points A, B, C. Determine the position of equilibrium. 

 (d). In connexion with the above the problem is directly associated with 

 the reciprocal iigures of Graphical Statics. 



The solution of the problem gives algebraic solutions of these and 

 similar problems. 



II. 



3. "We now take up the problem proposed and solve it, first, for the 

 plane, in detail along the lines of least resistance, subsequently giving the 

 method which admits of direct generalization to the higher spaces. 



