M 



.1 \ AL )'//< <, I oMETRY 



[Cii. II. 



If, instead of rectangular coordinate axes, oMi.jue axes 

 making an angle XOY=u> had been used, it would ha\. 

 been necessary merely to multiply the second nieml.. TS in 

 the results just found by sin &> in order to express the areas 

 of the triani: 



2. Polar coordinates. 

 Let the vertices <f the 

 triangle be P l =(p l , ft). 

 P., = (p* ft), and P, == 

 (p 3 , ft); to find its area 

 A iu terms of p,, p* p,, ft, 

 0* and ft. 



Pk-OPtP. 



>in rft-ft), 



sin (ft ft). 



Fio. 13. 



Manifestly, 



but 



A = OP,P 3 + 



and 



which may also be written 



A = J \ptfi sin (ft-ft^) + w, sin (ft - ft) 



+ W isin(ft-ft)j. . . . [5] 



The symmetry* in formulas [4], [4<i], and [/>] should be 

 carefully noted; it may be remarked also, that in the appli- 

 cation of these formulas to numerical examples, the resulting 

 areas will be positive or negative according to the relative 

 order in which the vertices are named. 



This kind of symmetry is known as cyclic (or circular) symmetry. If 



the numbers 1, 2, and 3 be arranged thus ii then the subscripts in the 



first term (in [4a] say) begin with 1 and follow the arrow heads around tin 

 circle (f.e. their order is 1, 2, 3), those of the second term bciiin with "2 aii'l 

 follow the arrow heads (their order is 2, 3, 1), and those of the third term 

 begin with 3 and follow the arrow heads. 



