PROJECTION IN TWO DIMENSIONS. 17 



Relation of Perimeters of the Primitive and its Projec- 

 trices. — In a former paper it was shown that if on a 

 primitive sohd we draw any arbitrary curve of length s, 

 and if Sj,, Sy, s^ denote the lengths of the curves passing 

 through the corresponding points of the projected solids, 

 then a simple relation can be found amongst the differ- 

 entials of these quantities. A. similar proposition holds 

 in geometry of two dimensions, the relation in this case 

 being between the perimeters of the primitive and its 

 projectrices. Differentiating (3) and (4) and squaring we 

 obtain 



dif = m^ [Idx + m dy) \ 



fi and 7;, being the corresponding points on the y-projec- 

 trixj we shall have 



^^ = l[[x-a)l^ {y-b)m), 

 whence 



dVi^ = dy^, 



d^j^ = r (/ dx -t- m dy) *. 



By addition we have 



d^^ + dr]'- + d^,^ + dr],^ = da:^ + dy"" + {ldx + mdy)\ (10) 



ds being the arc of the perimeter extending from the 

 point X, y to the point x + dx, y + dy, and ds^, dsy being 

 the arcs of the projectrices between corresponding points, 

 we shall have 



ds"- =dx^ +dy'-, 



ds/ = d^'- +d7j\ 



ds/ = d^,^ + dr]^^; 



also if (f> be the angle between the direction of the primi- 



SER. III. VOL. X. c 



