Fig. 1. 



5:2 M. P. Tchebychcf u/i a Modification of Watt's Parallelogram, 



which has with the desired motion eight, instead of five common 

 elements. 



We have done this, and fonnd that the approximation in 

 question may be attained by articulating, with each other and 

 with the beam, the four rods of Watt's parallelogram in the fol- 

 lowing manner. In this figure AB 

 represents the semi-beam upon which 

 it is required to construct a mechanism 

 capable of producing approximately 

 rectilineal motion along the vertical 

 line VV'j passing through the extre- 

 mity B of the beam when the latter 

 has a horizontal position. B C, D E, 

 C F, F G are the four rods composing 

 this mechanism ; C is the point whose 

 motion is to be considered; and F G, 

 turning around a fixed axis G, repre- 

 sents the counter-beam, as in Watt's 



parallelogram. These rods are articulated with each other and 

 with the beam in the same manner as in Watt's parallelogram, 

 with the sole difference that the rods D E and F C, instead of 

 being connected with each other, are articulated with the counter- 

 beam F G at two different points E and F. The lengths and 

 distances adopted are the following : — 



CF=FG= i/5 + 1 AB 3 



BD = EG= a/5 ~ 1 AB. 



Consequently BD is a mean proportional between AB and AD, 

 and EF is the half of AD. The rods BC and DE have the 

 same length; and provided the latter do not sensibly exceed the 

 semi-path of the point C, it may be arbitrarily chosen. The 

 centre of oscillation G of the counter-beam FG is chosen so that, 

 when the beam is horizontal, the rods BC and DE may be 

 vertical, and at the same time the rods Fig. 2. 



CF and FG may have the same hori- 

 zontal position, as seen in fig. 2. 



Such is the composition of the me- 

 chanism which, with the same number a D 



of rods as Watt employed, will give a 

 motion having eight elements in com- 

 mon with the desired rectilineal one. = - — 



This fact may be very easily verified by 

 determining, as a function of the incli- 

 nation of the beam, the variable distance 



