B, aPand bP in Fig. 109 these must by construction have a common normal passing through the instantaneous centre the describing line E F itself. Thus the two curves may serve as portions of profiles for the pair of elements which it is desired to construct. In the special case in which the primary centroids A and B are circular, the secondaries E and E are circles also, and the profile-curves a P and b P are involutes of those circles. This gives in spur-wheels the involute teeth which have been sometimes employed. Set-wheels can be made by making the angle F D constant for the whole series of wheels.

The profile-curves a P and b P are in this case roulettes obtained by rolling a straight line upon the two curves E and F. It must, however, be possible to describe them, as in the former case, as roulettes upon the primary centroids A and B. For circular cen- troids the auxiliary curve by rolling which upon A and B the involutes a P and b P can be respectively obtained is a logarithmic spiral.* If the middle curve of the three secondary centroids be not a straight line, the roulettes described by its points have not a common normal passing through the point of contact, and therefore are unsuitable for the profiles of elements.

### §34.

Fourth Method.—Point-paths of Elements used as Profiles.

The auxiliary centroids employed in the second of the methods which we have discussed may take the most various forms. A special case occurs when the auxiliary centroid coincides with one of the primaries. Here it no longer describes a curve in the latter, but each point in it describes there one other point only; relatively to the other centroid, however, it describes some point- path. If the latter be taken, as the profile of an element, the profile of the element with which it works must be a point. This method of constructing profiles has also been used for wheel-teeth. Fig. 110 is an illustration of the contact between teeth profiled in this

- This can be seen without difficulty. A demonstration is given (e.g.) by Willis

p. 92, another by Haton (Mecanismes). T>. 101.