ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 575 



extent connected with the action of force and the motion of bodies in 

 space, in which cases direction and position are direct factors in the 

 problem. In the great majority of cases the action of such forces may be 

 practically taken as in space of two dimensions, and therefore capable of 

 direct representation upon a plane surface ; but it is satisfactory to know 

 that the resources of graphic methods are equally capable of dealing with 

 cases in which three dimensions of space must be considered, and that, 

 by the aid of descriptive geometry, drawings on a plane surface may be 

 made to completely represent the conditions of such problems. Indeed, 

 the fundamental property which underlies the whole of graphical opera- 

 tions employed for the solution of problems, viz., ' reciprocity,' has very 

 similar interpretation, at any rate in mechanics, both in two dimensions of 

 space and in three. In any case, however, it is a plane surface which is 

 operated upon, and the question as to what can be done with a plane 

 surface as regards quantities represented or dealt with upon it must be 

 considered. 



Upon a plane surface we can represent the position of a point, and, 

 supposing the point to move, we can further represent its path by a line 

 or assemblage of points, any given portion of which has a definite 

 numerical length. In every position that the point takes along the line 

 it has a definite direction, and the direction of the motion of the point, 

 or, as it is commonly called, the ' direction ' of the line, for that position 

 of the point can also be stated as a definite numerical quantity. These 

 properties of direction and length, as already noted (vide second report), 

 are equivalent to a knowledge of the position of points on a surface. It 

 is these numerical properties which give the means of graphical operation, 

 because without such properties being introduced the operation would 

 be purely geometrical. Sometimes we may require as the solution of a 

 problem only one of these, as, for instance, length, in a mere problem of 

 graphical calculation. Sometimes another property may be required, e.g., 

 direction, as in the case of the line of pressure in an arch, or the slope 

 at any point of a bent girder ; but in all cases we employ two at least of 

 these properties in the graphical solution of a problem. 



It may scarcely be remarked that although we may actually operate 

 with straight lines we obtain by their intersections assemblages of 

 points giving curves of various orders, or, what amounts in the Umit to 

 the same thing, envelops, the straight lines being then tangents to the 

 curve. The distinction between these two ways of regarding curves, 

 although important in geometry, does not present itself very prominently 

 in graphical problems. 



The direction of a line at any point may be always supposed to be 

 about some centre, real or imaginary, to which it may be referred, no 

 matter what curve the line may take. In practice, however, the lines 

 nsed for graphical operations are chiefly limited to the case when the 

 centre is at infinity, that is, to straight lines. The reasons for this are 

 obvious, for straight lines of limited length or ' segments ' are readily 

 measured, and have a constant direction, which is easily expressed. Arcs 

 of circles in some cases might be employed, for the curvature is constant ; 

 but no curve offers the same advantages for the general purposes of 

 graphical manipulation as a straight line. There is another and deeper 

 reason for the use of straight lines or rows of points, viz., the fact that 

 forces act, and bodies tend to move in straight lines, and hence the 



