PROJECTION IN TWO DIMENSIONS. 11 



this axis to all points above ; hence, between corresponding 

 limits, we have 



Also if <f>{a^) be any arbitrary function of a?, we may show 

 in a similar manner that 



cr ff 



1 1 <t>{^) d^dr] = m^\\ 0(^) dx dy. 



JJo JJb (x-a) 



m 



As a particular example of the foregoing remarks, suppose 

 the primitive curve to be a circle of radius c, and suppose 

 the primitive axis to be a line through its centre ; then 



By substitution we obtain for the projectrix 



'm'^{cV—a)-\{y—lm{x — a)y = m'^c^. . . (6) 



To simplify this remove the origin to the point x=a, 

 y=o, and then refer it to new axes, so that 6, the angle 

 between the new and old axes of x, fulfils the following 

 condition : — 



tan 2^= -p ; (7) 



then the equation assumes the form 



.v^ y^ 



im'^& am^c"' _ ,^. 



I -i-m'' — v/(i +3»i^) (i — w^'•) i +m'- + ^{i +3^^) (i —m^) 



this represents an ellipse of which the area is mVc^. 



If we suppose the primitive circle to revolve round an 

 axis perpendicular to its plane, then m becomes a variable 

 quantity, and equation (6) will contain a single variable 



