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parallel to y. Initially 'i may be supposed to be an arbitrary function „f r wliile the faces of the plate, say at 0 and e, are maiiitamed at «nven temperatures. Ultimately the distribution of temperature is expres^d bv a linear function of r, say //' + Kz; and, as is known trom Fouribr's theory, the distribution at time t may be expressed by

f, — Jl' Kz + £ A„ e *■1

where u is an integer and p,„ dependiug also upon the conductivity, is proportional to After a moderate interval the terms eorresponding to the higher values of n becoine unimportant.

In the subsequent calculation it is eonvenient to take the ongin ot - in the iniddle surface, instead of as in (18) at one of the faces. Ihus

kz , 33-.- .

6 = //-f-Kz + Ale-^tcos — — Aie wcos — +. . .

— 4 e-vJgin"""- + A^-Msu^-

c 6

If 0' represent the value of fJ when reduced by the subtraction ot the proper linear terms as already explained, we find

r=Ate-r<'(co^-l)-A3e-^(co^+^) + .- ■

— A2e-'«(ti*^ — + Ae~'K' (i7/< V + ' ' ^0 >

After a moderate time the term in A{ usually acquires the preponderance, and then V — 0 «hen cos (k/c)= 2/sr. ^Vlleu t,le l)lale is looked at edgeways in the polariscope, dark bars are seen where 4- osOc, c being the whole thickness ot the plate.

As~a particular case of (19), (20) let us suppose tliat the distribution of temperature is symmetrical, or that K vanishes as well as the coefficients of even suffix A,, Ait H then represents the temperature at which the two faces are maintained, and (19) reduces to

0 = ƒƒ ! .cos — — A3ecos _ + • • • ...(21) c c

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