Heating and on Double Refraction resulting therefrom. 173 



the terms corresponding to the higher values of n become 

 unimportant. 



In the subsequent calculation it is convenient to t ike the 

 origin of z in the middle surface, instead of as in (18) at one 

 of the faces. Thus 



0=H + K* + A^-^'cos— - A 3 «-'»« cos — + . . . . 

 c c 



-A,*-** sin— +A 4 *-**sin^- (19) 



c c 



If d' represent the value of when reduced by the subtrac- 

 tion of the proper linear terms as already explained, we find 



6> = A x e^ (cos ~ - D-A.e-ft^cos ~ + ^) +. . . 



-A 3 ^(si^-£) + A^( s i a ^ + ^)-... (20) 



After a moderate time the term in A x usually acquires the 

 preponderance, and then 6 f = when cos (7rz/c) = 2/7r. When 

 the plate is looked at edgeways in the polariscope, dark bars 

 are seen where z= ±'2S0c, c being the whole thickness of the 

 plate. 



As a particular case of (19), (20) let us suppose that 

 the distribution of temperature is symmetrical, or that K 

 vanishes as well as the coefficients of even suffix A 2 , A 4 , &c. 

 H then represents the temperature at which the two faces are 

 maintained, and (19) reduces to 



= H + A^-'i'cos— -A 8 <rP»'cos— + ... . (21) 



c c 



If we suppose further that the initial temperature is uniform 

 and equal to ©, we find by Fourier's methods 



A^'tO-H), A 3 =A ( @_H), A s =^(0-H), ... 



and • '• (22) 



it Q' . / wz 2 \ . ■' / 3tt2 , 2 \ 



- ^ — cr = «-Pi' (cos — — - — I e~P*t I cos + s- 



4 ®— H V c it) 3 \ c SttJ 



+ le ^( C0S ^-D- (23) 



where also 



p,=d Pli p 6 =25 Ph &c. . . . (24) 



