Feb. 9, 1871] 
NATURE 
291 

itself, and this result agrees with the experiments. In | 
throwing weights with the arm Mr. Jevons found that there 
Aw (2w+ x)? 
wx Gw+x’ 
useful effect = 
(13) 

was no true maximum of useful effect, and that it increases | The condition necessary to make this expression a maxi- 
with the weight used. 
It can be shown from the equations used by me, which | 
are deduced from Law 1, that this result might have been | 
predicted from theory. | 
The useful effect is proportional to 
wx R, 
or to 
ZX oi Vi73 
which, by equation (2), becomes 
2 
w x7: ms 
this expression varies as the following, by equation (3) 
Zw 2) ya; 
Ca eee 
and since, by equation (1) or Law 1, 
w + +; we obtain finally 
wrx 

v? varies inversely as 
CuRVE OF USEFUL 
| mum is 
w(z2w+ x) (gw+7 x) 
(w+%)Bw+tr) 6w+x); 
which reduces to the quadratic equation 
4wv+3rw+x7=0 (14) 
This equation has imaginary roots, viz. : 
no) 
Ww 
Hence there exists no real value for the weight thrown 
which will make the useful effect a maximum, 
Mr. Jevons’ second set of experiments consisted in 
raising and lowering various weights by a pulley and 
cord, through the convenient range of the arm, and noting 
the number of times the weights were raised, the rapidity 
of the motion being maintained constant, 
HYPERBOLA 
BSC/SSA = WEIGHT 
RODINATE= USEFUL EFFECT 
| , 
M& Jevons’ SEXPERIMENTS, N° | 
THROW/NG WE/GHTS W/TH ARM 
The results of these experiments are 
Weight. No. of times raised. | 
56 Ibs. 5°7 
42 » II'9 | 
Fa Mee 230 
2I yy 37°6 
. I1IO’O 
TAs 35 Ae ceate 
It is easy to see, on theoretical grounds, that the weight | 
(x) of the arm will. disappear from the equations that re- | 
present the work done in these experiments ; for in raising | 
the weights the work done is proportional to 
(w — x) n, 
and in lowering the weights the work done is 
aN. 
These two, added together, give for the total 
Work done = w x. 
I have verified this anticipation of theory by introducing 
x into the Ze equations furnished by the #ve simultaneous 
values of wand #, and I find that the mean value of x 

turns out to be 
a = o'18lb., 
Ne IV. 

a value which, if the experiments were absolutely accu- 
rate, ought to become zero. Oh 
The rate of work is proportional to w, and multiplying 
this by the work done, in accordance with Law 2, we 
A (15) 
This equation gives the following values of A, correspond- 
jng to the five simultaneous values of w and 7 : 
| find 
wn 
A w 
17875 56 lbs. 
20992 TG eas 
TSO74 La is tet ra tose Fe 
LOGS Ziv adele. <a lh setae 2 EAs 
ZUG G6O\Sa <a) ict teh teitiem eGtess 
The mean of these values is 
A = 19017. 
Solving equation (15) for 7, we find 
A 
= 6) 
n= TA. (16) 
