218 
NATURE 
[JuLy 3, 1902 
the highest praise, for it possesses in a large degree just 
those qualities that one specially seeks in a work of this 
nature—qualities that will render it an indispensable 
addition to the library of every serious student of 
botany. Papers that have appeared in comparatively 
inaccessible or little-known periodicals are duly recorded 
in their places, whilst owing to the clearness of the main 
subdivisions and the excellence of the system of cross- 
reference, it is usually easy to search out all the 
literature cognate to any given subject. 
The translation of titles originally printed in unfamiliar 
languages is a useful feature that this volume will share 
with those dealing with other branches of science. 
Another character of special utility lies in the enumer- 
ation, under their appropriate subject-headings, of the 
new genera and species that have been published during 
the period covered by the volume. It is sincerely to be 
hoped that it will always be found practicable to continue 
to give such comflefe lists, although their preparation 
must necessarily involve no small amount of labour. 
It remains to be said that the typographical arrange- 
ments are clear and good, and the few printer’s errors 
on which we have lighted are so trifling as to be almost 
negligible. Those who have been concerned in its pro- 
duction are to be congratulated on the appearance of 
this, the first instalment of a great work the value and 
importance of which it would be impossible to overrate. 
J. B. FARMER. 
THE GEOMETRY OF COG-WHEELS. 
La Costruzione degli Ingranaggt. By Prof. D. Tessari 
Pp. xvi+226; with cight lithographed plates. (Turin: 
Fratelli Bocca, 1902.) 
HE study of the proper forms to assign to the teeth 
of cog-wheels in order that they may run smoothly 
affords such simple and useful illustrations of the prin- 
ciples of geometry of roulettes that it seems a pity that 
few mathematical students have time to interest them- 
selves in the mattter. In regard to the assertion of 
certain empiricists that even if the teeth are not con- 
structed on mathematical principles they will adjust 
themselves in the course of wear, it is pointed out that 
an immense amount of power will be wasted in wearing 
the wheels down, and instead of the teeth becoming 
adjusted they will run loose. 
Whatever form be assigned to the profile of the teeth 
of one wheel, it is possible to construct a suitable profile 
for the teeth of the second wheel, and the two profiles are 
said to be conjugate. The condition that two profiles 
may be conjugate is that they must both be roulettes 
traced by the same rolling curve on the so-called “ primi- 
tive” circles of the two wheels. If, however, a number 
of wheels are to be mutually interchangeable, the profiles 
of any two must be conjugate, and it is necessary that 
the generating rolling curve or “epicycle” should be the 
same for every wheel, and the most convenient form is the 
so-called epi-hypocycloidal form, in, which the portion of 
the profile outside the primitive circle is an epicycloid 
and that inside the primitive circle a hypocycloid. 
An inferior limit to the number of teeth is determined 
by the condition that contact between one pair must not 
cease until contact has taken place between the succeeding 
NO. 1705, VOL. 66] 
pair, and for smooth running it is further desirable for 
at least two pairs of teeth to be simultaneously in contact. 
To make the number of teeth a minimum, it is necessary 
to make the generating “epicycle” as large as possible, 
in which case the hypocycloidal form becomes a straight 
line; and it is shown by a diagram that the minimum 
number of teeth possible is nine for a pair of equal 
wheels, or six for a rack and pinion arrangement. 
Now if the teeth are cut radially down to the base, the 
narrowness of the base is a source of weakness, especially 
when the number of teeth is small and their height con- 
sequently considerable ; moreover, wheels of different 
sizes constructed in this way are not interchangeable. 
For such reasons as these another form is frequently 
adopted in which the profile is an involute of a circle, 
which is in this case of smaller radius than the primitive 
circle and is called the “ base circle.” 
We are next introduced to another form of gearing, in 
which the teeth of one wheel are replaced by circular 
cylindrical spindles, an arrangement which, by the way, 
was some years ago tried in the gearing of tricycles for 
the purpose of reducing friction, and is still illustrated by 
chain wheels. In this case the conjugate profile is a 
curve parallel to. an epicycloid, or in the case of interior 
cogs, a parallel to a hypocycloid. A particular case is 
that in which the ratio of angular velocities is as 1 to 2, 
when the spindles on the smaller wheel may be reduced 
to three or even two in number, and these work in 
rectilinear slots, an arrangement familiar in connection 
with the so-called “oval chuck” and the “trammel ” 
methods for describing ellipses. 
Hitherto the teeth considered have been of cylindrical 
form, with axes parallel to those of the wheels. We are 
now led to consider wheels with spiral teeth, an arrange- 
ment due originally to Hooke and White, and which is 
well illustrated by the diagonal rack and pinion coarse 
adjustment of the modern microscope. This arrange- 
ment has been supposed by some writers to eliminate 
nearly or quite all friction. Prof. Tessari, however, 
considers that this view is due to an erroneous opinion 
as to the nature of the contact existing between the 
surfaces, and, moreover, that further investigation from 
an experimental standpoint is desirable on the subject of 
whether any saving of friction is effected by helicoidal 
cogs as against cylindrical ones. Here is an important 
subject for researches which might well be carried out in 
a modern laboratory of experimental mechanics. 
In the eighth chapter a new subject is introduced, 
namely, gears for converting uniform into variable angular 
velocity, and the first point is the determination of the 
primitive lines of wheels adapted for the required pur- 
pose. In other words, we have to find the form of two 
perfectly rough curves which by rolling on each other 
about parallel axes will effect the required transformation. 
If we assume that the angular coordinates of the two 
wheels are 6 and 6’, and / is the distance apart of the 
axes, the polar equations of the primitive lines for any 
given relation between @ and @ are determined from the 
equations 7+7/= and rdd=7'dd'. Among the 
possible arrangements we note a pair of ellipses rotating 
about their foci, and combinations of elliptic, parabolic 
or hyperbolic arcs, also arcs of equiangular spirals ; 
many of these arrangements are illustrated by elegant 
Ve 
