on gbaphic methods in mechanical science. 585 



Addition of Segments. 



Concerning addition of segments, in the sense which it is suggested 

 that this operation should be regarded, only a few remarks need be made. 

 The chief points to be observed is that if a movement of a point in one 

 direction along a straight line is considered positive, movement in the 

 opposite direction is considered negative. Also, that if A, B, C, D, &c., 

 are collinear points, that is, points lying on a straight line, then 



AB + BC + CD + + MN + NA = ; 



which is the same as saying 



AB + BC + CD + + MN - AN" = 0. 



This rule of signs can be extended to curved lines and surfaces, and 

 English readers will find the subject treated at length in the first chapter 

 of Cremona's ' Graphical Calculus.' 



Combination of Segments. 



In the preceding paragraphs the direction of the segment has not 

 been taken into account as a measurable quantity, although the properties 

 of direction form the basis of certain constructions which have been 

 mentioned. It has been shown that the process of combination by drawing 

 equipollent (equal and parallel) segments is simple, and it need scarcely 

 be said that the order in which the segments are taken is a matter of 

 indifference, the resultant being the same in every case ; since no matter 

 in what order a series of movements are made the final change of position 

 must be the same, though this is generally stated and proved formally as 

 a soi't of proposition. Cremona further gives and quotes various other 

 propositions relating to the combination of segments which, though in- 

 teresting, have little bearing upon the subject matter of the present 

 report. 



One well-known property, however, of general application requires 

 notice. 



Let A B, B C, C D, D E, , . . fig. 7, be segments, the notation denot- 

 ing the direction in which they are measured. At any point O, in A B 

 draw two segments, O^a, Oib, of such length that, in the equipollent 

 diagram OA'B', A'B' becomes the resultant found by combining 0,a and 

 Ojt. Proceed in the same way with a point O2 in B C, which is found by 

 producing O1& to meet BC, the triangle O B' C being the equipollent 

 diagram. It is clear that the order of procedure might have been re- 

 versed, and any point being first chosen the rays OA', OB', . . . might 

 have been drawn, and the polygon Oi O2 O3 . . . then found. 



These two diagrams have the well-known relation, that for every 

 point from which lines radiate in one there exists a corresponding closed 

 figure in the other. 



Thus to Oi corresponds the triangle O A' B' 



02 „ „ OB'C 



03 „ „ OC'D' 

 • • • » 11 ... 



If the end lines O^a and 046 be produced to meet at O5, then to O 

 corresponds the polygon O, O2 O3 O4 O5. 



