404 Proceedings of tJte Royal Society of Victoria. 



Fiff.i 



It will be found that all the components (i.e. 3rd, 5th, etc., in 

 this case) are the sides of isosceles triangles passing through the 

 origin, and that the altitudes are 



— ///3^, /i/o\ —/i/7'\ etc. respectively. (See Fig. 1.). 

 (The same can be quickly arrived at geometi'ically). 

 Hence, if the full wave or Cj be represented by 



Ci = ^?iSin(w/— ^]) -(-^3sin3(o>/ — ^.,) + rt5sin5(w/— ^g) + etc., 

 its third component Cj is 



= — — [rtisin(3a)^— ^i)+rtoSin3(3w/— ^3) 4-a5sin5(3a)/— ^5) 

 o~ 



+ etc.], 

 and its fifth component C^ is 



-^;^[ffisin(5w/ — ^i + a3sin3(")w/— ^3) + a,sin5(5o)/ — ^,,)+etc.] 

 " 



and so on, but by definition C., and C-, are also given by 



C8=a3sin3(<o/— ^3) + <;,isin9(w/- 6'y) + 



C5 = «gsin5(w/— ^5) + «igSinlo(a)/— ^jj) + 

 hence, identifying the expressions for the same components, we 

 find that 



^1= — 3'^(73=5^«5= — 7^«7 = etc,, 



^1=3^3== 5^3=7^,=etc., 



