592
�NOTES.
�to simplify, that of the twist pure and simple, which for every reason it
appears to me better to treat as the ultimate and general case.]
13 (pp. 64, 119 and 121). [It must, I think, be admitted that it is very desirable to have a somewhat clearer understanding as to the use of these names than has hitherto existed. I shall be very glad if the following table assist in any way in promoting this understanding. It shows the nomencla- ture I have myself used throughout the book, which differs somewhat from Professor Keuleaux's, but which agrees with that adopted by our best writers on the subject, so far as I have been able to make out a system from their references to these curves. The last column requires a word of explanation, the rest'of the table, 1 think, explains itself. The case referred to there will be understood by a reference to Fig. 97. Two circles are there in internal con- tact, of which the larger rolls and the smaller is stationary. It may seem at first sight unnecessary to separate this case, for what I have called the peri- cycloid can always be described as an epicycloid. The necessity for having a separate name arises, however, from the fact that the curves described by points without and within the large rolling circle, that is the curtate and prolate peri- trochoids, cannot be described as curtate or prolate epi-trochoids. The cardioid can be generated as a roulette as a special case both of the epi- and the peri- cycloid. I have retained " curtate " and " prolate " for want of better words, but they are sometimes singularly inappropriate to the external form of the curve, in Fig. 96 for example, where the larger ellipse is the curtate curve. Professor Cayley's kru-nodal and ac-nodal hardly seem adapted for popular use, and an excellent suggestion of Professor Clifford's, " looped " and " wavy," fails also in (external) suitability in the case of Fig. 96.
I use trochoid as the general name for the whole class of curves.]
�DESCRIBING
POINT.
TROCHOID,
EXTERNAL CONTACT.
INTERNAL CONTACT.
CIRCLE ROLLING
UPON
STRAIGHT LINE
(LINEAR TROCHOIDS.)
CIRCLE ROLLING
UPON CIRCLE.
(EPI-TROCHOIDS. )
SMALLER CIRCLE,
ROLLING.
(HYPO-TROCHOIDS.)
GREATER CIRCLE,
ROLLING
(PERI-TROCHOIDS. )
On circle
Beyond circle
Within circle
Cycloid.
Curtate trochoid
Prolate trochoid.
Epicycloid.
Curtate epitro-
choid.
Prolate epitro-
choid.
Hvpocycloid.
Curtate hypotro-
choid.
Prolate hypotro-
choid.
Peri cycloid.
Curtate peritro-
choid.
Prolate peritro-
choid.
�14 (P. 119.) Chasles (Aperq u historique sur VOrigine des Methodeb
en Geometric, 1837), cites Cardano's Opus Novum de Proportionibus Numerorum Motuum, etc.
15 (P. 121.*) Without the use of a model it is very difficult for anyone unaccustomed to this class of problems to realize fully the nature of the motion which occurs here. In my model the curve-triangle, UTQ, is etched upon a glass disc, the duangle, PVQW, is engraved upon the plate KPSQ. [Professor
- The reference is to the top line of p. 121, the 5 of the reference number has unfortunately
been omitted in printing.
�
