164 
.AIE. J. II. JEAXS OX THE VIBRATIOXS AXD 
We can calculate the value of the integrals which occur in this expression, and the 
sum of the first two terms inside the curledjjrackets is found to be 
2,1 + 1 ^ 
Hence we may write (I U) in the form 
w 
here 
(G = Ch’ (xr) -f- [k<.() 
_ A + 2 /i, ^ 47rp 1 
irp jrK^ K~ 
(•24), 
D = - 
Airp 
a 
-n + k 
(2,1 + 1)2)-K 
e now have, from equation (16), 
■iirp d 
d 
^ ~ dx + d7 
(2/( + 1) p~ dx 1 
+ fo 
and hence 
f/r ^" + ( 2 ;i+ l)y^ 
(25). 
(26). 
§ 1-j. llieie aie thiee boundary-conditions to be satisfied, expressing tliat the 
normal pressure and the two tangential tractions shall vanish at every point of the 
fiee surface. As LawB“' shows, these may he represented by three symmetrical 
equations, to be satisfied at the surface v cy, each of the tvpe 
^ ('’'0 + d (= 0 . 
hiil/stituting for ^ and u I'rom (25) and (26) this becomes 
d 
XA.r -p jx 
h/ / dddi ' d d 
.faC' .,r 
iirp (2n — 2) d 
+ ^ , 0 , 
(2,1 A 1) 2-r dx a" 1 
’r"a S 
+ /Y- 
dx 
dx 'dr ~ 
— 0 
(27). 
§ 13. Write 
d 
dx 
so that the right-hand members are solid harmonics of deo-rees ii - 1 and -(n A- -H • 
then ^ - V -r -A- 
* L.VJli:, lor. rit. aide, j). 191. 
