THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 135 
m >I: 
Qar fe oy ee G = ) 
ite = ga Inbp cos mw B Ima” \ BR a a7 Bad, Bat mJ,,Ba 
+ gl bate Ba +m— i= 33 ,Ba) | f > p< a 
2a a” cos Mw 
ey : 2mp™ (pt = =| MI PA ae Bal,’Ba) + dm+ sete Ba —m+ + 23,4) Hs pra. 
The corresponding integrals with log R in place of {R* log R—+R’ may be obtained at 
once by taking vy’ of the above. 
Also, all through we may write log (R/c), log (p/c), log (a/c), instead of log R, log p, 
log a, this amounting merely to a change in the unit of length. 
(g) By equating coefficients of like powers of 8 in the results of (c), (f) we can 
obtain 
i Ve GoxRp”*"cos mw'p'dp'dw 
and i ie (4R? log R ~ £R2)p’m+2n cos mo'p'dp'da’. 
In the case when m=n=0 
ay, 2 2 5 
ioe Gy«kRp'dp'dw’ = — eas TT yKpKaG kK’, p<G 
ojo K? 
K2 
2 
= = 7) Gykpkad 9 Ka > pra. 
he | ae log R - 7B! \pdpde' = = { p* + 4p?a?(2 log a — 1) + a4(4 log a — 5) \ >» pe 
4 
= ( Fptlog p = iP? )ara? + log p™e » pra. 
These results and those of (f) may easily be verified, or obtained, from the values 
: : hs dl dl : 
of y* of the integral, with the conditions that I, ee wel a are continuous at p=a. 
(i) In certain problems a class of potential functions occurs, which may be deduced 
from the fundamental potential 1/r, where r°?=a*+y’+2’, by successive integration 
with respect to z. 
Writing 
log (r +2) 
log (r+2)—-r 
" a 
2(@ — 3p") log (r +2) — Gra + 3p” 
I} 1 
we may easily verify that u,, w., uv, are potential functions, and that 
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