498 Mr. O'BRIENS CONTRIBUTIONS TOWARDS A SYSTEM 



4. When we say that a symbol, o for instance, represents a straight line, we mean that a 

 defines the magnitude and direction of the line, but not its beginning; in other words, the line is 

 supposed to be drawn of a given length and in a given direction, but not from a given point. 



However, if the contrary be not specified or implied, we shall always suppose the line to begin 

 at the origin, i. e. at a certain point chosen for the purpose of reference. 



5. We shall use the term Direction Unit to denote a straight line of a unity of length drawn 

 in any particular direction. We shall always use the letters a, ft, y to denote direction units, and, 

 unless the contrary be stated, we shall also suppose these three directions to be at right angles to 

 each other : in other words, we shall assume a, ft, y to represent three straight lines drawn at right 

 angles to each other, and each a unity of length. 



6. We shall divide symbols into two classes, Number Symbols and Line Symbols, the former 

 representing numerical quantities positive or negative, the latter straight lines in magnitude and 

 direction. 



7- We shall define the jiosition of a point in space by the Line Symbol representing its 

 distance from the origin : thus, whenever we speak of the point a, we mean the point whose 

 distance from the origin is represented in magnitude and direction by the symbol a. 



In our idea of distance here we suppose direction, as well as magnitude, to be included. 



8. If a, b, c be any line symbols, it follows, from the first principle above stated, that 

 a + h + c represents the distance of the end of the line c from the begitining of the line a ; the end 

 of a being supposed to coincide with the beginning of b, and the end of ft with the beginning of c. 



In like manner a — b denotes the distance of the end of a from the end of ft, a and 6 being sup- 

 posed to have the same beginning. 



Hence, if a and 6 be the symbols of any two points A and B, a - ft is the symbol of the 

 right line drawn from B to A, and ft - a the symbol of the line drawn from A to B. 



9. If X be any number symbol, and a any direction unit, xa represents a straight line of the 

 length a: drawn in the direction a. 



Hence, if r be the length of a right line drawn from the origin, a; y z the lengths of the 

 co-ordinates of the end of that line, and a ft y the direction units of the three co-ordinate axes, the 

 three co-ordinates will be represented by the symbols xu, yft, ssy, and the line by the symbol 



ma + yft + zy. 



This symbol also defines the position of the point whose co-ordinates are oc y z. 



If a ft c be the direction cosines of the line, its symbol becomes 



r (aa + hft + cy). 



The coefficient of r is evidently the direction unit of the line. 



10. Let r and r' be the lengths of any two lines AP and AP" drawn p 



from a point A, and let e and e represent their direction units; then the symbols ^^___- j 



of these lines will be re and r e, and therefore the symbol of the line PP' will be * •" 



re — 1 e- 



If r = r and e' - e is indefinitely small, this expression becomes 



rde. 



Now in this case PP' is at right angles to AP, and therefore it follows that rde is the symbol 

 of an indefinitely small line perpendicular to the line re. 



