BE, EYEEETT ON THE EIGIDITY OE IEON AND COPPEE. 
365 
Let j6 (as in last paper) denote the quotient of one of the distances (2) or (3) accord- 
ing as we are dealing with torsion or flexure, by the distance (1). Then since (3 is the 
tangent of a small angle (which call B), we have 
& tan B=(l+tan 2 B)SB=(1+|3 2 )£B, 
and therefore the angle turned by mirror (c$B) is found by dividing the observed deflec- 
tion (tS tan B) by 1+/3 2 , or, what amounts to the same thing, multiplying it by 1 — j3 2 . 
Let B n /3 1 and B 2 , (3 2 be the values of B, (3 for the nearer and further mirrors respect- 
ively; then a tan B 2 -A tan B 1 = (1 +/3 2 )SB 2 — (l+|3 2 )c)Bi, which may be shown to be 
equal to 
m 2 and m l being any two numbers which have the ratio § tan B 2 : h tan B,. The observed 
values of T and F are proportional to £ tan B 2 — o tan B„ and must therefore be multiplied 
by the correcting factor 1- 
which, when j3, and /3 2 are equal, becomes simply 
1-/3 2 . 
In the present series of experiments (3 1 and (3 2 w T ere sensibly equal, their common value 
being about ‘069 for torsion and ‘064 for flexure. By squaring these numbers we 
obtain the corrections 
-•0047 T and --0041 F, 
which are applicable not only to the experiments on the malleable iron rod, but also 
to those on the cast-iron and copper rods. 
Adding together the mechanical and optical corrections, we obtain the total corrections 
+ •0027 T and + -0017 F. 
T T 
The resulting correction for p is +• 001 |f, which applied to the values 1’272, 1‘278, 
1*273 gives, after subtracting unity, -273, *279, -274 as the corrected determinations of 
the .value of <r, and we adopt the mean of these, which is -275. 
We proceed to calculate the torsional and flexural rigidities t and f. 
The mean values of T and F, uncorrected, are respectively 210-7 and 165-1, which, 
when corrected as above, become 211-3 and 165-4. 
Now we have 
t— twice distance X force X arm X length X -t-T, 
f— twice distance X force X arm X length X * JLy F, 
where, in centimetres and grammes weight, we have twice distance =445 -6, force=100, 
length=29-45, arm=55-71 for torsion and 55*83 for flexure. 
Hence 
log £=9-71893— log T— 7-39403, 
%/= 9-71823 -log F=7-49969. 
The rod having been cut at the places where the mirrors were attached, the central 
3 e 2 
