10 Rev. 0. Fisher on the Surface Elevations and other 



Now the fundamental partial differential equation for the 

 conduction of heat gives 



dv = d*v 

 dt ~ da?' 



Making this substitution, the above becomes 



3e C r , .„ cPv , > , ,(Pv 



(^?I (r -* ) -^ < *-*> d?- 



If we integrate by parts, so as to raise the index of (r— .r) 2 , 



observing that -yji may De P u ^ = when x = r, we can 



obtain Mr. Davison's expression No. 2 in the 'Proceedings' 



of the Royal Society*. 



Putting the members of the above as an equalit} r , the value 



of x will give the depth of the level of no strain. 



d 2 v 

 If we integrate y-r 2 first, we get by parts 



Y -— C r V - t2 



= -{r-x) 2 —L=e ***+ 2(r-x)— -=€ u, d.r, 



\TTKt ) x \1TKt 



because 7 



-- = when x = r. 

 dx 



But 



(Pv V 



dx* ^TTKt 



_i? ( 2.v \ 



—V . . , 



Hence, dividing by , transposing, and writing «.- for 



\TTKt 



4/ct, we have for the equation to find x, 

 (73^p|0— *)*« aJ -\ 2{r-x)e «'dxj=(r-x)e 



(r — x)' 2 

 Multiply by v [ , and transpose. 



(r-x) 2 Jt 

 ST" 6 "" 



a? C r 2* -*2\ . (r-xf -I2x 

 2rJ x \ a- J 0/- or 



= 1/ '"'"•• 



* Vol. xlii. p. 826. 



