54 M. P. Tchebychef on a Modification of Watt's Parallelogram. 



vertical motion, we have only considered how many elements 

 were common to the vertical and to the curve described by the 

 point C ; the degree of approximation of the two, however, 

 depends also very essentially upon the position of these elements. 

 We have already examined this question in the first part of a 

 memoir entitled " Theorie des mecanismes connus sous le nom 

 des parallelogramnies " *, and therein proposed methods for 

 rendering such an approximation as perfect as possible. By 

 applying these methods to the case under consideration, we 

 should be led to introduce certain small changes in the values 

 of the several parameters, in order to render the mechanism as 

 perfect as possible. By means of these corrections, the devia- 

 tions of the point C would be reduced in the proportion of about 

 1 to 2 7 (see § 5 of the memoir cited). But since, in practical 

 cases, these deviations, as we have seen, are themselves very 

 small — amounting at most to some hundredths of a millimetre, 

 —it is evident that by the application of the above corrections 

 the theoretical precision of the apparatus might be carried to a 

 limit unattainable by the mechanician. For ordinary practical 

 purposes, therefore, there is no inducement to seek a mecha- 

 nism capable of giving rectilineal motion with greater accuracy. 

 This improved mechanism is the more worthy of attention, since, 

 as we have shown, the utmost desirable precision is attained by 

 employing the same number of pieces as in Watt's parallelo- 

 gram, whose practical defects are often experienced. 



We may observe, lastly, that the adjoining new form of the 

 mechanism is obtained by changing the signs of the radicals in 

 the foregoing values of the elements. 



In this new form, where Fig- 3. 



^-UB, 



CF = FG= 



BD=EG 



4 

 ^5 + 1 



AB, 



the degree of working precision is 

 theoretically the same as before; its 

 construction, however, would neces- 

 sitate the prolongation of the beam, 

 and thus be attended with great dis- 

 advantages. 



* Mtinoires de Savants Etrangers, vol. vii. 



