According to differential geometry and gear geometry, the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion. The correct meshing condition, contact line, contact ratio, calculating method for pin tooth's maximum contact point are developed. Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1, 2, 3 and 1, respectively. A general method called enveloping method to generate hypocycloid and epicycloid is put forward. The correct meshing condition for cycloid pin wheel gearing is provided, and the contact line and the contact ratio are also discussed.pin wheel, cycloid, conjugated profile, equation of meshing, enveloping method, contact ratio, contact line Cycloid drives are widely used in many industries, such as machinery, mine, metallurgy, chemical, textile, national defense, etc., due to their large gear ratio, compact size, high load capacity and high efficiency. More attentions are paid to this drive in precision transmission because half of its teeth are meshing simultaneously, because the outstanding error average effect leads to high precision, and because it has high torsion stiffness for there is no flexible element. Additionally, cycloidal gear pumps based on the principle of cycloidal pin drives are attached importance in many countries due to their smooth transmission, low pulsation and low noise.Generally, the gear geometry of cycloidal drives is described as: curtate cycloid by outer and inner rolling method; curtate epicycloid and pin teeth satisfying the Willis law; terms of continuous transmission [1,2] . Compared with involute gears, the meshing principle of cycloid drives has the following defects. 1) Lack of close math deriving. There are no descriptions for meshing equation, meshing line, etc., which greatly interrelate with the transmission traits. 2) Meshing theory being not systematic. For example, planetary transmissions for one tooth difference and two teeth difference are discussed separately [3] ; the essence of planetary gear conjugated profile is not revealed when the inner gear is given as pin wheel. 3) Existing contradictory conclusions. For example, the term of continuous transmission is stated as that pin wheel should have one more tooth than cyc-
