to the case of an Autotomic Plane Circuit. 



451 



paragraph, the author enters upon the first section of his sub- 

 ject, viz. : — 



Descriptio figures per motum recta parallelum. 



At the outset two conventions (or lemmas, as they are called) 

 are adopted, which may be paraphrased as follows :— 



1. A straight line, whether positive or negative, having par- 

 allel motion and moviDg first in one direction and then in the 

 opposite, thereby describes parts of the same figure which are 

 affected with opposite signs. 



2. A straight line which, during motion in one direction, is 

 at one time positive and at another negative, thereby describes 

 parts of the same figure which are affected with opposite signs. 



A circuit resembling the lemniscate in the character of its 

 area is then considered. The describing line begins at g 

 (fig. 1), where it has no magnitude, moves parallel to itself from 

 g to c, varying the while in 

 length, changes sign in pass- 

 ing through c, and, then pro- 

 ceeding as before, at last va- 

 nishes and ceases to move at 

 /. The area of the figure 

 g a c efd cbg is therefore, in 

 accordance with the second 

 convention, to be reckoned as 

 the sum of two areas, g acbg 

 and cefdc } affected with op- 

 posite signs. 



As one might without spe- 

 cial notice consider the point 

 c in this figure either as a 

 mere meeting-point of two 

 portions of the perimeter, or 

 as a point of actual intersec- _ 

 tion, the resultant area being & 



of course different in the two cases, the author is here led to 

 remark on the importance of indicating in the diagram the 

 mode of tracing the circuit. This, he says, can best be done 

 by traversing the perimeter and bordering it on the right with 

 one colour and on the left with another ; so that at points 

 where the opposite vertical angles have the same colour, the 

 perimeter merely meets itself; and where they have different 

 colours, actual intersection takes place. The colours also render 

 important help in determining the sign which affects any parti- 

 cular portion of the figure. On this principle the author's 

 diagrams are drawn ; but instead of two colours, one on the 



2G2 



