357 
of Edinburgh , Session 1873-74. 
and are reversible face to face.” This theorem is general so far 
as geometry is concerned, but is restricted in its mechanical appli- 
cation by the condition of material continuitj 7- , and hence there 
arise some difficult and interesting problems. 
Each assumed form of undulation has its peculiar path for the 
point of contact ; this path is obtained by drawing normals PX, 
meeting the line of abscissae in X, as in figs. 1 and 9, and then 
through some fixed point Q, technically called the pitch-point , 
drawing QP' equal and parallel to NP. 
If we suppose the point P, in fig. 1, to move steadily along the 
curve EXTYY, accompanied by the normal, the point N will move 
continuously, though not uniformly, from E to Y ; and from no 
point in the axis ESTUV can more than one normal be drawn to 
the curve. Such a rack would give rise to a system of wheels 
having only one point of contact, and so useless in machinery. 
On augmenting the ordinates HP in some fixed ratio, we obtain a 
curve with deeper undulations, and augment the subnormals HX 
in the duplicate ratio ; in this way we may cause the point X to 
pass beyond S before P has arrived at X. In such case N must 
become stationary, and then return to S, when P shall reach X ; 
passing back towards E, N must again become stationary, and 
thereafter progressive, reaching T just when P does so. In this 
way the motion of N along EY will resemble the direct and retro- 
grade movements in longitude of the superior planets. Parts of 
the line of abscissae will thus be traversed thrice, once forward, 
once backward, and once forward again ; and from each point of 
these parts three normals may be drawn to the curve. Ey properly 
adapting the ratio of enlargement we may cause the first stationary 
position of N to be at T, and then every part of the pitch-line is 
traversed thrice, — that is to say, wheels deduced from such a rack 
must always touch in three points. If we augment the ordinates 
in such a ratio as to bring the first stationary position of X onwards 
to U, every part of the pitch-line will be traversed five times, and 
the corresponding wheels will always touch at five places. In this 
way, when the general character or equation of the curve is deter- 
mined on, we can discover the exact depth of tooth giving any 
specified odd number of contacts. 
In machinery we should have at least two teeth completely engaged, 
