408 Proceedings of tlte Royal Society of Victoria. 



which is 



sin-^sin3(a)/ - /x/2) 



=r — [sin^sin(w/'-/x/2) + — +etc.] 



IT 2 3" 



COS— cos 3(a)/ - /a/2) 

 + -[cos^cos(w/— /a/2)+--- ^^ +etc.] 



TV Z o 



where mix=zH{TT — ix)^=h. 



When /A= 120° the above expression for the triangle reduces 

 to 



i/l ( r, J COs9o)/ COS 15a)/ , , 1 



+ ^-A cos3a)/ + — -- - -\ — — + etc. - 



dir^ \ 6' 0' ) 



where tan/3=z7 — ^■ 



12((r), Wave form a polygon with // vertices per half wave 

 and such that the functions of its vertices are all equal and also 

 the projections of its sides on the axis of x. 



J^et q=:m;^ — w.j = Wo — W3 = ;;/g — m^ = etc.:=//i„-\-f/ii and let the 

 abscissae of its vertices be a, a + Trjn, a + 'Iir/fi, . . a + {/i — Ijir/n 

 then by § 11 the expression for the wave is 



'^^S . N(«/ + 7r/2-[a + ;-7r/;/]) 



■i 



where r has all values from o to // — 1. 



Substituting for the N functions their equivalent harmonic 

 series, summing the terms that have the same arguments and 

 remembering that n(/^2f;i^, the exj^ression for the polygonal wave 

 under consideration becomes 



. sin(oj/- a + 7r/2//) -f ^ sin3(a)/— a+ 7r/2//) 



Htt I • TT oo • 3?? 



^sin — O'sm 



-} sui(oj/ - a-^-Ttjln) -\- 



3-sin 



-| ^— sin5(a)/— a + 7r/2;;)+ etc. }- 



\1{d). If in example {c) 7i become infinite the polygon becomes 

 a smooth curve satisfying the following conditions 



rL ff 



; const. 



dx- 



