170 Prof. T. A. Hearson. [May 30, 



No advantage is" derived from analysing a machine into parts such 

 as pairs if it requires the whole machine to complete one of the 

 parts. 



The enunciation and the explanation of the influence of the first 

 law previously mentioned, of the constancy of the sum of the four 

 angles of a quadrilateral in governing the association of the Oil and 

 I motions in one mechanism, is one of the important original features 

 of this paper. 



The influence of the second law, viz., that the two sides of a 

 triangle are together not less than the third, in limiting the associa- 

 tion of the O, U, I motions, is now also for the first time pointed out, 

 though Reuleaux and others have, without formally enunciating the 

 law, made use of the fact to determine the proportions necessary for 

 certain suggested movements. 



By the application of these governing laws one is able to write 

 an exhaustive list of all the possible combinations in one simple 

 mechanism of the three simple 0, U, I motions, and to explain why 

 other combinations are precluded. 



Fourteen distinct combinations are possible, and only fourteen. 

 They are exhibited by the following formulae, in which a large O 

 associated with a small o signifies that in one case adjacent links 

 turn relatively to one another so as to continuously increase the 

 angle between them, and in the other to continuously diminish the 

 angle. The double signifies that two complete revolutions accom- 

 pany one complete to and fro swing or slide. 



Group. 

 Group. 



C Group. 



Following Reuleaux, the author applies the principle of the " in- 

 version of the kinematic chain," considering it to be a continuous 

 sequence of links in a closed circuit containing a sequence of 

 elementary motions. In explaining what is meant by inversion, it is 

 pointed out that relatively to an observer or user of a machine one 

 piece is fixed. This is called the frame of the machine. Each one of 

 the four links may in turn be made the fixed or frame link, and 



