242 
Mii. L. X. G. ^ILU^" ON THE ELASTIC EGUlLIBEirM OE 
the croitis-sections, v;//‘ heiiig constant for the section. Straiglit radii therefore remain 
straig’ht radii. 
Further, r(f) = [r-ja-) (its value at the boundaiy). 
In other words, the transverse shearing-stress across cylinders coaxial with the 
given one is zero for sections where there is no such external ajjplied stress, and 
foi' other sections diminishes rajjidly along the radius as we go inwards, so that near 
the centre it is always small compai-ed with 
There is one very important point to be noted with regard to this method t)f 
approximation : p and a increase with a, and tlieretore, however small ay may be, so 
long as it remains finite, we still reach a value of a, for which it is not justifiable f<-> 
take for and the hrst terms of their expansions in positive integral powers of the 
argument. If, however, we stop at the rth term, where v is finite, then if I!^ is the 
remainder after v terms of the series 
cos 
2/j -f Ittz 
for example. 
and if, on tlie other hand, the numerical value of the difference 
•f^' __ h (p) 
ko? I. i'x) 
< e for all 
values of n not greafer than v, whei'e e is a quantity which depends upon c/o, and 
wliich can be made as small as we please by making cia small enough, then the 
difierence 
^ h(p) 'In l-nz 
^ Q„. V— 7 “ COS - 
2c 
V (« 4 r 
0 /,; «- 
COS 
2/1 -f lirz 
2r 
must be numeiically less than 
1-h I + 111/1 + 
re 
where j x \ deiujtes the numerical value oi' modulus of x, and 11/ 
r terms of tlie series 
c„ 4 /' 
V Jl 
cos 
2 /t -I- 1 77 ^ 
a~ 
is the remainder alter 
iSow if both the (/liginal series and the approximate series are uniformly 
convergent, then by giving r a certain value, jll,, j and lll/i can both be made less 
tlian a certain small quantity 77/0 which tends to zero when v tends to infinity, and 
that for all values of r. 
Now make c/o so small that e < r^/Sr, whicli we can always do so long as r is finite. 
I'hen the difierence between the two series is numerically <" t), and the approximation 
holds. 
If, however, for any value of it becomes impossible to assign an upper limit to 
li„ or Iv/, i.c,, if either series cease to be uniformly convergent, then we should luive 
to increa.se v indefinitely in order to make Hi,, j < t;/ 3 , and therefore to modify e, so 
