406 ProceedhH/s of the Royal Society of Victoria. 

 y = i/tx -\- K, we see that 



/l^= //l^ + M, «2 = *'^2 + ^5 '^3 == '^'-ci + ^^i ^h ^= 'f'-A — M, 



lience 



«3 — ;/./ = 2M 



n^ — ?/^^vi.^ — ;;/, 



fli — ^'r, '= >"i — »h,- 



Thus, if we call ;ii^ — m.^, the function of the vertex ^c^.^, we see 

 that by addition of / to the given wave a new vertex is intro- 

 duced whose function is equal to that of / ( = 2M) while the 

 functions of all the other vertices are unchanged. It is easy to 

 see that the function of the vertex of — /is — 2M so that if / 

 be subtiacted from the given wave a new vertex is introduced 

 whose function is — 2M. 



If the abscissa X of the vertex of / correspond with that of 

 one of the vertices of the given polygon x^^ say, then no new 

 vertex will be introduced by the addition (or subtraction) 

 of /and 



«1 — ;72=:Wi — ;//2 



«2 — «3= W2 ~ ^'h + 2M 



«3— «4 = '''"3 — '''4 

 «4— «5 = ^'^4 — Wfi 



If in addition 2M = Wg— ;//2> ^hen «2— "3 = fi-nd the vertex or 

 break at x^^ is removed. Thus, by subtracting from the 

 polygonal wave an isosceles wave whose vertex has the same 

 function and abscissa as a vertex of the given polygon, this 

 vertex of the polygonal wave is removed, while the functions of 

 its remaining vertices are unchanged. To remove each vertex 

 therefore a definite isosceles wave is required, and since, when 

 all vertices are removed the axis of abscissae orjj' = remains, 

 we see that the sum of the several isosceles waves required 

 to extinguish the given wave is equal to the latter. 



In the general case, therefore, of a polygonal wave with ti 

 vertices, the vertex w,., w,+i, Xr,r+\ will be removed by 

 subtracting the isosceles wave /:=AN(w/ — a) where 

 m r — w,.+i = 2M = ihJTr 

 Xr, r+1 = X = a + 7r/2, 



