M. P. Teh eby chef on a Modification of Watt's Parallelogram. 53 



of the point C from the vertical line VV (fig. 1) *. It will 

 then be at once seen that the vertical VV is a tangent, at 

 the point corresponding to the horizontal position of the beam, to 

 the curve described by C ; further, that in the neighbourhood of 

 this point the curve has seven elements in common with its 

 tangent, and, lastly, that it cuts the same at a distance from G 

 less than BC ; so that, within the space described by C, the curve 

 and the vertical have necessarily an eighth common element. 



We see from this with what extreme rapidity the deviations of 

 the point C from the vertical line VV (fig. 1) increase with the 

 amplitude of the oscillations of the beam, the distances being of 

 the seventh order with respect to the inclination of the beam. In 

 ordinary practical cases, where the inclination is never great, the 

 working of this mechanism would, as far as precision is con- 

 cerned, be greatly superior to that of Watt. Take, as an ex- 

 ample, the case treated by Prony in his well-known " Note sur le 

 parallelogramrae du balancier dela machine a feu "f, where the 

 length of the semi-beam AB is 2'515 metres, that of the rod 

 B C being 0*762 of a metre, and the greatest inclination of the 

 beam 17° 35' 30". With the improved mechanism the devia- 

 tions from the vertical would be less than 0*05 of a millimetre, 

 whereas, according to Prony, the deviations with Watt's paral- 

 lelogram would amount to 2 millimetres, — a quantity forty times 

 the above, and far from being insignificant in the working of a 

 machine of this description. 



Hitherto, in seeking to approach as closely as possible to a 



* This variable distance is expressed by the formula 



— - — . AB (cos ^— cos <£), 



wherein the angles f and $ are functions of the inclination oc of the beam 

 which satisfy the two equations 



A 3-V5 V5-1 .V 

 ^1-— ^— cos* ^__cOS(£j 



+ ^ AB ~Y~ Sm * + ~~2~ Sm *) ~~ AB* 

 (l-cos.+ ^cos^-^cos^) 2 



lAB" Sm ^ + ~4^ Sm</)+ ~T _Sin V~"AB 2 * 



The approximate expression for the distance in question is consequently 

 given by the series 



7-3V5 AB 2 V5-2 AB 8 



W~ BC* + — I6~BC 2 * + '" 



f Annates des Mines, vol. xii. 



