S6 LORD EAYLEIGH. 



parallel to ^. Iiiitially ô may be supposed to be au arbit]'ary functioii 

 of z, M'iiile tlie faces of the plate, say at and c, are maintained at 

 giveu températures. Ultimately the distribution of température is expres- 

 sed by a linear function of r, say //' -|- Kz; and, as is known froin 

 FouRiEii's theory, the distribution at tirae f may be expressed by 



Ô = H' + Kz + S .-/„ e -l'n' sln , (IH) 



c 



where n is an integer and p,), depending also upon the conductivity, is 

 proportional to rr. After a moderate interval the terms corresponding 

 to the higher values of ti becoine unimportant. 



In the subséquent calculation it is convenient to take the origin of 

 c in the middle surface, instead of as in (18) at one of tlie faces. Thus 



Ù = HA- Kz -\- A. e-i'^f cos — — A., e-i'-'' cas ^^ -f . . . 



e ' c 



, . '^TTZ , , . . 4^7rz ,,„. 



— A., e '"''-' s m \- yLe~>'^hm (19) 



c c 



If ô' represent the value of ô wlien reduced by the subtraction of the 

 proper linear terms as already explaiued, we find 



f/' '^^ ^\ . f / •i'^- , '^ \ , 



ô=A,e-^'^'{cos^--)-A,e-"^'^[cos — ^-—) + .. . 



-Me-"4.i..^-^ + A,e-"^^(eiu^ + ^-. . . (20) 



\ c TTC/ \ C ZTTC/ 



After a moderate time the term in Ai usually acquires tlie prépon- 

 dérance, and theu (5' = when cos (tt^^/c) = S/tt. When the plate 

 is looked at edgeways in the polariscope, dark bars are seeu where 

 z= ± ■280c, c beiug the whole thickness of the plate. 



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

 of température is syminetrical, or that K vanishes as well as the coefti- 

 cients of even suffix A.^, yl;^, S)-c. ^then represents the température at 

 which the two faces are maintained, and (19) reduces to 



Ù=H-\- A.e~i''fcos — — yLe-''''coi-—-\- (21) 



c c 



