Aug. 18, 1879] 
Substituting for A and x inthis equation their values above 
given, we can obtain by calculation the distances to which 
the weights should be thrown, according to Law 1. 
We thus obtain the following comparison between 
theory and observation, 
w R (observed). R (calculated). Difference. 
56 lbs. 1°84 ft. 1°99 ft. —oro6 ft. 
28 ,, 3°70 5 351 55 +0'19 ,, 
DAS 55 6°86 ,, Gobr5, & . 0°80 ,, 
foes TO:S OM mit; STOO2) =). -t-Or54) 
a ye Jorma slbeiies RO Dag cea SoA wep 
2» - . 1865 ,, IQ'Il ,, —o'46 ,, 
I » ey 2305 53, 4 23°85 ee —o'80 ” 
2 ” 4 7A syn fa 20°39) 53) —0'24 ;, 
The agreement here shown between observation and 
calculation founded on Law 1, is quite as complete as the 
agreement between observation and the empirical for- 
mula used by Mr. Jevons, which may be written, in the 
notation of the present paper, as follows :— 
(2w-+ 7°8) R= 2313. (6) 
Mr. Jevons’ third series of experiments consisted in 
holding various weights on 1'c hand extended horizon- 
tally, and noting the time duiin which the weights could 
be so held. The following are the weights and times 
observed :— 
w a 
18 lbs. 14°8 secs. 
14 5 3275 
10 ” 60°3 ” 
7 » 87°45 
Ae Deitel. site MALO) 55, 
PER gabe Sets ae ORS ee 
I ” 321-2 ” 
Omitting the first of these experiments I find that 
Law 2 satisfactorily accounts for the remaining six, and 
gives a constant, which is nearly identical with that ob- 
tained from my own experiments made in 1863. 
When the arm is extended horizontally, if allowed to 
fall through an indefinitely small arc, the centre of oscil- 
lation falls like a free body under the influence of gravity, 
and the muscles then lift back the arm through the same 
arc, and this goes on continuously until the muscles are 
tired out. 
Let us use the following notation :— 
w and x are, as before, the weight held in the hand and 
the weight of the arm. 
Z = radius of oscillation ; 
a = distance of centre of gravity of loaded arm from 
centre of shoulder joint ; 
és = small space through which the centre of oscilla- 
tion falls; 
n* = number of such falls during 
¢ = whole time required to fatigue the muscles. 
The total work done by the muscles in the time /, is 
evidently 
(w +2) 4 ns; 
but, 6s varies as /, and, therefore, the total work done 
varies as 
(w+ x) : ze 
The rade of work is evidently proportional to 
(w+ x) = 
and since, by Law 2, the total work done before fatigue 
multiplied by the rate of work is constant, we obtain 
(w +x) | t = Const. (7) 
*I have ascertained the number » from acoustical observations made on 
the muscular swsurrus. 
NATURE 425 
And, since 
a@_3 (2w + x)? , (8) 
7 *@w +2) Gw+) 
we find, by substitution, 
(2w+ x)4 Bs 
OEE t=a, (9) 
This equation (9) is the statement of Law 2, as 
applied to Mr. Jevons’ experiments ; and we are required 
to find values for x and a, which wiil make equation (9) 
best correspond with the given observations. 
I find, by trial, that the following values will answer 
best :— 
AnD: 
a@ = 22,050. 
If we solve equation (9) for ¢, we find 
f—— (3 nine ee ( 10) 
(2w-+ x)4 
From this equation, substituting the values of x and 4, 
we obtain the following comparison of observation and 
theory : 
w z (observed). ¢ (calculated), Difference. 
14 lbs, 32°5 SECS. 34°2 secs. — 17 secs 
10 5, 60°35, 54°75 +56 , 
7 » 87°45 84°38, +26 , 
4» 1479 5 1476 ,, =e OSes 
2 » 2189 ” 234°4 ” —15°5 ” 
I ” 3212 ” 305°5 ” -F15°7. ” 
This comparison is very satisfactory, the differences being 
much less than possible errors of observations. Mr, 
Jevons’ experiments further show that the wseful effect 
has a maximum corresponding to a certain weight. This 
weight, which gives the maximum of useful effect, may be 
readily calculated from Law 2. 
By equation (10), the useful effect is 
wi= A _w3wt 2) 
(11) 
2w-+ x) 
This will be a maximum, when 
(aw +2) Qw+ 4) =8w(3w +2); 
or when 
6u? — 34 Ww — F#—O0; 
or when 
ww ——— (12) 
or, 
WwW = 0°73 x. 
Substituting for x its value 7'4lb., we find for the weight 
that gives the maximum useful effect, 
w = 5‘4olb. 
The useful effect observed by Mr. Jevons was as follows : 
w Use'ul effect 
18 lbs. . 266 
Tf. sy, Cee eerie wh 45S 
10); See ss) go ct | 003 
se 2 emer ree ts Ole 
7 re OS. Poe neta Mee 
Pte hs. fell cu an erature Zlefe) 
es ete cre Sel! 
The actual maximum corresponds to 5"4lb. lying between 
7lb. and 4lb. 
I may observe, in conclusion, that the difference of 
weights + of the arm, found in the two sets of experiments 
is quite natural. 
In the experiments in which the arm was held out 
horizontally, its weight, 7:4lb., is the weight of the arm 
below the centre of the shoulder joint. 
Inthe experiments in which the weights are thrown by 
the arm, a portion of the shoulder blade is in motion, in 
addition to the simple arm, and the total weight becomes 
81lb. SAMUEL HaUGHTON 
