ANALYSIS OF THE KINEMATICS OF MECHANISMS. 517 



directions of the normals to the touching surfaces cannot be very accurately 

 obtained on the drawing unless the " pitch points " T AB , T BC , &c, are known, and 

 if these are known to start with, the various velocities can most simply be 

 determined from them directly without reference to the touching points. 



The point T AB may be looked on as indicating two points, one in the first 

 wheel, which may be called T A , and the other in the second, which will 

 be called T B . To obtain the velocity of T A the triangle pat a is constructed 

 similar to the triangle P A AT A . In this triangle at a coincides with the line ab l} 

 and pt a is perpendicular to P A P B . Making a similar construction for the 

 velocity of the point T B , we find that the point t b coincides with the point t a . 

 Thus the points T A and T B in the two wheels have the same velocity pt ab , and 

 the point T AB is therefore called the "pitch point." The angular velocities of 

 the two wheels are therefore inversely as the distances P A T A and P B T B , this 

 being a familiar theorem proved in the ordinary treatment of toothed gearing. 

 Similarly, if pt bc be drawn perpendicular to the centre line P B P C to its inter- 

 section with b. 2 c, this pt bc is the velocity of the pitch point T BC of the pair of 

 wheels (BC). If the teeth be so shaped as to give constant angular velocity 

 ratios between the wheels, the points T AB , T BC , &c, in the diagram of the 

 mechanism and the points t ab , t bc , &c, in the velocity diagram remain fixed 

 throughout the periodic motion of the train. It may also be noticed that since 

 at a ._ AT A , Va _. B/T B tWpforp at a pt b . . aU _ AT^ P B T B PbTb 



P t a ~ ivr A a ph - Ivr B ' lliereiore ^ a • ij b - biU - P A T A • E,T B ~ P A T A ' 



cit 

 so that ^y a ^ so measures the ratio of the angular velocity of wheel A to that 



of wheel B. The condition that the angular velocity ratio should remain 

 constant may thus also be expressed by the condition that the line ab in the 

 velocity diagram, representing the velocity of sliding of tooth over tooth, should 

 be divided in a constant ratio by the fixed point t ab . [This point t ab is only 

 fixed if the angular velocities themselves, as well as their ratio remain constant, 

 the magnitudes of these angular velocities being proportional to pt al .~\ Whether 

 this proposition can be utilised in simplifying the practical drawing out of the 

 teeth-profiles, so as to secure a constant velocity ratio, the author has not yet 

 had time to investigate. 



Velocity and acceleration diagrams have been completely worked out for the 

 Joy's valve gears used by Mr F. W. Webb on his compound locomotive engines, 

 the gear being differently arranged for the high and low pressure cylinders. 

 These mechanisms are too complicated to be treated without very inaccurate 

 approximation by any other graphic or algebraic process known to the 

 author. 



