292 
NATURE 

Substituting for A and w, in this equation, we find the 
following comparison of theory and observation : | 
w n (observed). n (calculated). Diference. 
56 lbs. Cee nr. = OO — 03 (45) 
He es 119 JOS. ee + 11 (7x) | 
2B. ap te sete he OMe cen eA — 12 (zy) 
2) irc COMDEV (by Be 4371 — JSG an 
sO eis Oh ne cae! OKS) O72 +13'0 (4) | 
This table is satisfactory, as the differences are less than | 
possible errors of observation. 
The wseful effect admits of no maximum in these ex- | 
periments ; for it is represented by wz, and 
wxun=A; i 
therefore— 
(17) 
which represents an equilateral hyperbola ; zw becoming | 
infinite when the weight is zero, 
The agreement between Laws 1 and 2 and Mr. Jevons’ | 
experiments, may be best shown by means of the accom- | 
panying tables and diagrams, which represent the curves of | 
useful effect, calculated from Laws 1 and 2. | 
| 
First SET OF EXPERIMENTS. 
Throwing weights—It follows from equation (5) that 
the useful effect 
rs A 
wn = —}; 
w 
wie wa) 
w+x gut ap 
Substituting in this equation, the values 
A = 262'2 
z= $81 
we obtain the following table :— 
=wR=A 
(18) 
USEFUL EFFECT No, 1. 
w w JR (observed). aw J (calculated), 
oO oO oO 
2 bia ey WR Re pe S38) 
I BEG ia | eae tO, 
2 Fhe eats 3 Oia a te aie Poe Ste 
4 FE CLONE Ns 556 
tg td eT a a 70'! 
Ty Nigel dG 2 GOAL sa el ween 160'4. 
Ze ee ee ROO ma TeeOo 
S| ghar ie eo EOL CRE yr . 106°5 
In the accompanying curve No. 1, the abscissa is the 
weight, and the ordinate the useful effect ; and the centres 
of the small circles represent the actual observations, 
SECOND SET OF EXPERIMENTS. 
Litting weight with pulley and cord.—\n these experi- 
ments the useful effect wz may be at once calculated | from which we obtain four equations to determine the 
from equation (17), from which the following comparison 
is made, and Diagram No. 2 constructed. 
Substituting for A, its value 19,017, we find— 
USEFUL EFFECT NO. 2 
w zn (observed), aun (calculated). 
14 lbs. 1554 <=. AS5D 
PH ny allah leks OO tase 905 
200. O44) 2s. . G79 
Dein hah. 8 x OOD emre 453 
oy arge abs ae 319. 340 
The curve represented in Diagram 2 is a portion of an 
equilateral hyperbola, whose abscissa is the weight, and 
its ordinate the useful effect. 
THIRD SET OF EXPERIMENTS. 
Holding weights on hand extended horizontally.—The 
useful effect in these experiments may be calculated 
from equation (11). 

Useful effect= we = 4 @ (34 +4) 
wae) 
Substituting for 4 and x their values 
A = 22050 
x= 7'4 
we obtain the following comparison, and construct Dia- 
gram No. 3. 
USEFUL EFFECT No, 3. 
w zut (observed). qw# (calculated), 
fo) OU. ter ane ° 
I « -92tt ee Sano 
2 <a oes ee meena 
Pre ey See 2) ic 590 
7 A. Gl a, EMO ene ane 594 
TOW: Meas Gosh ak 547 
TA ie 6 BOS Ne re 479 
18 266 421 
In Diagram No. 3, as before, the abscissa of the curve 
represents the weight, and the ordinate denotes the 
useful effect. 
The equations of the three curves which represent the - 
useful effect in Mr. Jevons’ three sets of experiments, 
when expressed in Cartesian co-ordinates, are as follows :— 
x (2x+ a)? 
yo Aner a) Geta ae 
ytd No. 2 
x 3x + @) 
1.8 A epee an 
It will be interesting, inconclusion, to explain why the 
simpler empirical formula, used by Mr. Jevons, coincides 
so nearly with the more complex formula deduced from 
theory. 
According to Mr. Jevons’ empirical formula, the useful 
work done in throwing weights is 
Bw 
2w +x 
where B = 2314 ; and, according to my formula, deduced 
from theory, the useful work is 
Aw (2w + x)? 
wa) Gw + = 
where A = 262°2. 
These two expressions, algebraically considered, can 
never become identical, but may become nearly so, if 
(w + x) (Bw + 2)? = K (2w + 4)3, (19) 
in which & is a co-efficient nearly constant. 
Expanding both sides we have 
g9wi+i5xrw?+7a2°w+t v3 
=K (8wi+ i2zxrw?+627w+ 2); 
best value cf 4, viz. : 
9= 8kK K=T1'125 
I5~=12K K=1250 
7= 6K K=1166 
1= K K=r000 
Mean . .. 1135 

The co-efficients of the two sides of equation (19) will 
be most nearly equal, when A = 1°135. 
The constants used by Mr. Jevons and myself give 
K=4 226225 
which is about the value required by the preceding con- 
siderations. 
The curve denoted by Mr. Jevons’ equation is an 
equilateral hyperbola, the co-ordinates being parallel to 
fae ia oat and the origin taken upon the cur ve 
itself, ae 
[ Fed. 9, 1871. 
