Hannonie Analipls. 409 



^r=Wj when x=o and =^ — Wj when „v=:7r, 



v=o when x:=<> and wlien .v=— . 

 This curve is the parabola 

 w, „ 



whose axis is .v=:7r/2 and vertex .v=:7r/2, j'=:/;/j7r/4, 



and the harmonic expression for the wave of wliicli it is the 



type is obtained by niakino; ;;=:oo and a=zo in the expression in 



(c) and is 



8wi f . ^ , sinSoj/ sinSo)/ , ^ 1 

 ^ { sinoj/ + - - -\ _— + etc. - 



13. To find the harmonic expression for the complete jDeriodic 

 function whose graph for one ]ieriod is made up of the sides of 

 two equal and similar triangles ABC and A'BC so placed that 

 A' and C lie in AB and CB produced respectively. Take A. as 

 origin, then the abscissae of B and A' will be tt and 'Iir respect- 

 ively and let the abscissa of C:=/a hence that of C' = 27r — /x. 



By geometrical construction the different components of this 

 wave can be easily obtained if we remember formula 1 § 1. 



Thus to get the 2nd component we cut the wave in four 

 portions by ordinates at 77-/2, tt, 37r/2, 27r, invert the second and 

 fouith portions, superpose them and the thiid portion on the 

 first, add the corresponding ordinates, divide each sum by four 

 and the plot of the results will be a half wave, which gives the 

 2nd component. 



It will be found for the wave under consideration that all the 

 components are, in general, trapeziums of the type treated in 

 §12 (a); and if the trapezium which is the rth component be 

 specified as in § 12 (a) by w,- and jj.,. measured on the original 

 scale of abscissae, it will be found that 

 , tanA + tanB 



(i.e. that '2r times the function of its vertex is equal to + the 

 function of the vertex of the triangle) 

 and that 



sinriMi — +sinr/u, 

 the same signs being taken together. 



