172 Mr. Tovey on the Elliptical Polar kation 



hence, if we suppose s^ = s',, the two values of p will be p, 



/ s s' 

 = — 1 and p.2 = 1 ; and, assuming * / ' ,'„ to be a very 



small quantity, we shall have s'm b = 1 and b = - , nearly. 



27r 

 Now, since k = , where A is the length of a wave, if 



A 



these values be substituted in (18.), and the terms involving 

 Ay , A'a , be comparatively insensible, there will result 



.-A B 2'^ ^'^'^ 



Since -j- =z v, and k = , the expressions (2.) may 



be changed to 



Yj = a sin < - — {vt — x) > , 



^ = p a sin < {vt — x)— b v . 



Now either of the values of v, and the corresponding value 

 of ^, may be substituted for v and g in these expressions. But 

 since the equations (1.) are of the first degree, they may be 

 satisfied not only by the values of ») and ^ corresponding to 

 each value of v, but by taking for r) and ? the sums of these 

 particular values, in which we may change the value of a 

 as V changes. Hence the equations (1.) may be satisfied by 



rtz=ajsm-l~{v,t-x)\' + a^ sin<j^ -^(i^^^— a?) | ^^g.) 

 ll=z Pi a^sinS ~{v,t-x)-b\ + p2a^s\n-l^{Vot-x)-b\. 



If in these expressions we give to p,, poi and b, the values 

 just assigned to them, namely, —1, 1, and — , we shall have 



>j = a, sin J -^ {Vft — a-) I +00 sin -l -r— (^^2^— ^) f 



r 2 TT , ,1 / 2 TT , , ,"1 (^^0 



^ = — ttyCOS*! {v, t—x) ^+C2C0S \-~r-'{v^t — X) > . 



Now suppose a quartz crystal to have two parallel faces 

 perpendicular to its axis. Take x in the direction of the 



