66 



NATURE 



[May 24, 1877 



are pivoted, the word " linkage " is employed to denote any 

 combination of pieces pivoted together. When such a 

 combination is pivoted in any way to a fixed base, the 

 motion of points on it not being necessarily confined to 

 fixed paths, the link structure is called a " linkwork : " a 

 "linkwork" in which the motion of every point is in some 

 definite path being, as before stated, termed a " link- 

 motion." I shall only add to these expressions two more : 

 the point of a link motion which describes any curve is 

 called a "graph," the curve being called a "gram." 



Fi^. 1. 



The consideration of the various properties of these 

 " linkages " has occupied much attention of late years 

 among mathematicians, and is a subject of much com- 

 plexity and difficulty. With the purely mathematical 

 side of the question 1 do not, however, propose to deal to- 

 day, as we shall have quite enough to do if we confine 

 our attention to the practical results which mathemati- 

 cians have obtained, and which I believe only mathemati- 

 cians could have obtained. That these results are valuable 

 cannot, I think, be doubted, though it may well be that 

 their great beauty has led some to attribute to them an 

 importance which they do not really possess ; and it 

 may be that fifty years ago they would have had a value 

 which, through the great improvements that modern 

 mechanicians have effected in the production of true 

 planes, rulers and other exact mechanical structures, can- 

 not now be ascribed to them. But linkages have not at 

 present, I think, been sufficiently put before the mecha- 

 nician to enable us to say what value should really be set 

 upon them. 



The practical results obtained by the use of linkages 

 are but few in number, and are closely connected with the 

 problem of "straight-line motion," having in fact been 

 discovered during the investigation of that problem, and 

 I shall be naturally led to consider them if I make 

 "straight- line motion" the backbone of my lecture. 

 Before, however, plunging into the midst of these link- 

 ages it will be useful to know how we can practically 

 construct such models as we require ; and here is one of 

 the great advantages of our subject — we can get our 



results visibly before us so very easily. Pins for fixed 

 pivots, cards for hnks, string or cotton for the other pivots, 

 and a dinmg-room table, or a drawing board if the former 

 be thought objectionable, for a fixed bise, are all we require. 

 If something more artistic be preferred, the plan adopted 

 in the models exhibited by me in the Loan Collection can 

 be employed. The models were constructed by my 



brother, Mr. H. R. Kempe, in the following way. The 

 bases are thin deal boards painted black ; the links are 

 neatly shaped out of thick cardboard (it is hard work 

 making them, you have to sharpen your knife about every 

 ten mmutes, as the cardboard turns the edge very 

 rapidly) ; the pivots are little rivets made of catgut, the 



Fig. 3. 



heads being formed by pressing the face of a heated steel 

 chisel on the ends of the gut alter it i^ passed through 

 the holes in the links ; this gives a very firm and smoothly 

 working joint. More durable links may be made of tin- 

 plate ; the pivot-holes must in this case be punched, and 

 the eyelets used by bootmakers for laced boots employed 

 as pivots ; you can get the proper tools at a trifling 

 expense at any large tool shop. 



Now, as I have said, the curves described by the 

 various points on these link-motions are in general very 

 complex. But they are not necessarily so. By properly 

 choosing the distances at our disposal we can make them 

 very simple. But can we go to the fullest extent of 

 simplicity and get a point on one of them moving accu- 

 rately in a straight Ime ? That is what we are going to 

 investigate. 



To solve the problem with our single link is clearly 

 impossible : all the points on it describe circles. We 

 must therefore go to the next simple case — our three-link 

 motion. In this case you will see that we have at our 

 disposal the distance between the fixed pivots, the dis- 

 tances between the pivots on the radial hnks, the distance 

 between the pivots on the traversing link, and the dis- 

 tances of the tracer from those pivots ; in all six different 



distances. Can we choose those distances so that our 

 tracing-point shall move in a straight line ? 



The first person who investigated this was that great 

 man James Watt. " Watt's Parallel Motion," invented in 

 17X4, is well known to every engineer, and is employed in 

 neatly every beam-engine. The apparatus reduced to its 

 simplest form is shown in Fig. 2, 



