MOORE. — MINIMUM GEOMETRY. 213 



The element of arc in this system of coordinates is, 



ds = xdy — ydx, 



and likewise the angle between two nearby tangents of a curve is 



da = udv — vdu, 



where the curve is expressed in angle coordinates. 



Using the definition for area as was used in Case I, it was shown in 

 A. P. G. that here also we have two areas: 



Line area = a + b + c = Abe = Cab = Bac. 

 Angle area = A + B + C =? ABc = ACb = BCa. 



From these formulas it is at once seen that the line area of any closed 

 curve is equal to its length. That is for a closed curve, 



Line area = Length = f {xdy— ydx). 

 If the curve is given in angle coordinates, 



Angle area = Angle sum = J~ c (udv — vdu). 

 If the equation of a curve is written in the form, 



y = f 0), 



the tangent at the point (xi,y\) is, 



xdy — ydx = X\dy — y x dx or 



dy dx 

 ds y ds 



If x and y are functions of s the tangent at a nearby point is 



or (dy d?y \ (dx , d?x \ 1 



The angle between these two tangents is, 



j _. (dy_ d 2 x dx d 2 y\ 

 \ds ds 2 ds ds 2 ) 



The angle area then becomes, 



(7) Angle area = Angle sum = J(| * - | g) d, 



