152 
5IK. L, X. G. FILOX OX THE ELASTIC EQUILIBRICM OF 
§ 3. General Solution for a Symmetrical Strain. 
Let r, (f), z be the usual cylindrical co-ordinates ; also, following the notation of Tod- 
HUXTER and Pearsox’s ‘ History of Elasticity,’ let st denote the stress, parallel to els, 
across an element of surface perpendicular to dt, .s, t standing for any t^^'o of the 
letters r, <f), z. 
Let R, u: denote the radial, cross-radial, and longitudinal displacements respec¬ 
tively, then we have (Lame, ‘ Lecons sur fElasticite’), if u, v, iv are independent 
of (f) : 
,, , d~H . , (I / /'. \ (l~l( ", . , it-ic 
+ -d) + -V) p, (7 ) + + (^ + /") = 
7 7~ir ) + A.,. = 0 
di 
dr 
(/-A 
+ f) 
(t-i/, 1 di>.\ 
dnh 
'ddir . 1 did 
+ ,■ , 7 ; ) + '■* (,>,■= + Jr ] + (^ + ® 
(!)• 
(^)- 
(3). 
rr _= (X -f Lg) + X ^_ + X 
fd/ic . dll \ d 
= 
dr 
dr 
dz 
^dz 
(-i), 
dz 
(dr V 
^[fr-; 
X and jx 1)eing the elastic constants of Lame. 
We see from the above that and ref) depend only on v, the other stresses only 
on u and w. xLlso the ecpiation (2) contains v only, (1) and (3) contain u and w only. 
The solution for transverse displacements is therefore absolutely independent of the 
solution for radial and longitudinal displacements. 
Let us IKnv denote the operators - f r by and , bv D. Ditferentiate (l) 
dr r dr " dz ^ 
with regard to 5 . and (3) Avith regard to r, and remember that the order of tlie symbols 
1) and -S' is indifferent, then we find 
