Permanent Forms of Mathematical Expressions. 12 1 



ments at the point, in the secondary wave which represents 

 the component sina along the axis of z, are, 



U'= /^,r,0-0sina 

 V = - ^( x, r, - x Jsina 



W = <p( x, r,d—y Jsina. 



Now consider a new set of axes y and z such that the 

 new axis of y is coincident with the direction of the original 

 displacement. The change in the co-ordinates of the point 

 considered will simply be that we have — a instead of 0. 

 Then, by the principle of superposition, we have 



U + U' =f(x, r, - a), 

 (V +■ V')cosa + (W + W')sina = <p(x, r, - a), 

 (W + W')cosa - (V + V')sina = \P(x, r, - a), 



or 

 f(x, r, 0)cosa +flx, r, - „ Jsina =f(x, r, - a) . (1) 



(p(x, r, 0)cos 2 a + </>( x, r,d--z Jsin 2 a 



+ -< \p(x,r, 0) - \p( x, r, - - J j- sinacosa, 



= (p(x, r,d-a) . . (2) 



4>(x, r, 0)cos 2 a + \p( x, r, - ^ Jsin 2 a 



- < <p(x, r,6)-<p(x,?;d--y) j- sinacosa 



= ^(x,r,6-a). . . . (3) 



Since the left hand side of each equation is to be a function 

 of (0 — 0) we conclude 



f{x, r, 0) =/iCos0 +/ 2 sin0 ~\ 



<p(x, r, e) = <p!- ip 2 sin0cos0 + 2 sin 2 J- . (4) 



\p(x, r, 0) = fa + i/> 2 cos 2 - ^ 2 sin0cos0. J 



