568 
Transactions of the Boyal Society of Sonth Africa. 
the spring does not increase the equivalent mass of the apparatus by M. To 
show this we obtain a further approximation to the equivalent mass when 
— is small — equal to e, say. 
cot 
^ gives 6=^—X where \ is small. 
tan X \ 
appro X. whence A= 
2 ande=^_2 
4m M 
= -.-5 roughly. 
Thus for small additions the equivalent mass of the spring diminishes by 
about one-fifth of the mass added. 
^5 4. To show how m' varies between its extreme values a graph has been 
drawn with ^ as abscissa and 
of the graph was to tabulate values of 
as ordinate ; the method of construction 
cot 
and of ^o— --^-^ for values 
of 0 from 0° to 90^ at intervals of 5°. The line sliowino- the value — 
^ m 3 
has also been drawn in the same figure to show the deviation of the 
approximate result for M large from the true result as M varies. 
For reference the tabulated values are oiven here. 
0 
cot 9 
1 cote 
9 
not 0 
1 cot 0 
9 
0° . 
00 
. -3333 
50° 
•9616 
. ^3515 
5 . 
130-9788 
. -3335 
55 
•7295 
. ^3555 
10 : 
32-4940 
. -3340 
60 
•5516 
. -3604 
15 . 
14-2553 
. -3349 
65 
4111 
. ^3659 
20 . 
7-8710 
. -3363 
70 
•2980 
. -3722 
25 . 
4-9149 
•3375 
75 
•2047 
. -3793 
30 . 
3-308 
. -3396 
80 . 
•1263 
. -3866 
35 . 
2-3378 
. -3418 
85 
•0590 
. -3954 
40 
1-7071 
. -3448 
90 
0 
. ^4054 
45 . 
1-2732 
. -3479 
§ 5. To test the theory experiments were carried out on a spiral spring 
of mass 164 grammes, length 38 cms., and diameter 4 cms., made of steel 
wire of circular section 1*35 mm. diameter. The statical value of K (force 
necessary to produce unit extension) was found to be 16^96 grammes' 
