Abstract: According to the princinciple of double-enveloping, The author pointed out that a plane tooth-protile has "two" conjugate curves and derived a general epuation for the second onjugate curve. The author also proposed a new method for the calculation of engagement of cycloidal gears whose rolling circles coincide with the pitch circles of the conjugate gears. The nain poin- of the mathod aces the first conjugate curve degenerates into a point, the tooth-porfile of the conjugate gear is therefors determined by the second conjugate cnrve. A simple equation for determining conjugate tooth-profils and two examples using this equation are given in this paper.