160] THE GEOMETRY OF THE CYLINDRO1D. 153 



while, if 6 be the angle XOT, and OT be denoted by r, we have 



x = r sin(0 - a), 



or x = y cos a x sin a (ii) ; 



but, obviously, 



OR = r cos (0 - a) + M T ; 



whence h = x cos a 4- y sin a + z cot /3 (iii). 



Solving the equations (i), (ii), (iii), we obtain 



x e = x sin a + y cos /3 cos a, 

 y = + x cos a + y cos /3 sin a, 

 z = h tan /3 y sin /3. 

 It appears from these that 



# 2 + y z = x* + ifcos 2 ft, 



a; ?/ = #?/ cos /3 cos 2 a + (y 2 cos 2 /3 # 2 ) sin a cos a. 

 The equation of the cylindroid gives 



z (x z + y 2 ) = Zmx y ; 



whence we deduce, as the required equation of the section, 

 (h tan ft - y sin /3) (# 2 + y z cos 2 /3) 



= 2m3ry cos ft cos 2a + 2m sin a cos a (y 2 cos 2 ft a?); 

 or, arranging the terms, 



sin /3 cos 3 fty 3 + sin yS?/a; 2 (m sin 2a + h tan /3) # 2 + Zmxy cos /3 cos 2 



+ (m sin 2a cos 2 ft h sin /3 cos /3) y J = 0. 



It is often convenient to use the expressions 



x = h tan (6 0) m sin 20 cot ft tan (0 a), 

 y = hsecft msin 20cosec/3, 



from which, if be eliminated, the same equation for the cubic is obtained ; 

 or, still more concisely, we may write 



x = y cos ft tan (0 a), 

 y = h sec ft m sin 20 cosec ft. 

 This cubic has one real asymptote, the equation of which is 



ysinft = m sin 2a + h tan ft, 

 and the asymptote cuts the curve in the finite point for which 



x = tan 2a (h + m sin 2a cot ft). 

 The value of at this point is a. 



