PROJECTION IN TWO DIMENSIONS. 7 



If AC were projected on Oij, the length of the projection 

 would be mhC ; from the point D draw a perpendicular 

 such that 



'D¥=mAC, (2) 



and complete the parallelogram. Multiplying together 

 (i) and (2), we get 



DE.DF=m^AB.AC; 



AB . AC is the area of the primitive parallelogram^ and 

 FD . DE is the area of the parallelogram FDE. By pro- 

 jecting on the line O]/, we may obtaiuj in a similar manner, 

 another parallelogram such that 



NK.KL = /^AB.AC. 



Hence A^ and A^ denoting the projected areas, we have 



A^-l-A^=A/^ + Am% 



for 



r + m^ = I . 



If the rectangle CAB have any motion of translation, this 

 will affect the positions, but not the magnitudes, of the 

 projections ; if the rectangle have a motion of rotation 

 round any axis perpendicular to its plane_, each projection 

 will vary in magnitude, but their sum will be constant. 



The reasoning of the above simple case may be extended 

 to any plane area bounded by curved lines ; for we may 

 suppose the area to be rigidly connected with two straight 

 lines on its plane, and at right angles; then the whole 

 area may be considered as the limit of a series of elemen- 

 tary parallelograms whose sides are parallel to these axes. 

 If a denote the area of one of these elements, its projection 

 Uj^ on a line parallel to the axis of a; will be m^a, and 

 summing, we have 



