Regarding the utilization of the energy of propeller race by Contra-Rudder to the propulsive perfomance, the author cleared the currrent-pattern of propeller-race and handled this problem on the basis of the lifting line theory, to examine, from the viewpoint of velocity field of propeller-race, the propriety of the propeller theory that has hitherto been advocated. This report deals only with the fundamental theoretical calculations and three important results induced therefrom ; and the contrasting of the fundamental calculations with the experimental results produced by various counting methods will be described in a succeeding report. The items which were cleared up in this theoretical counting, and the matters wherein the errors in the theory hitherto advocated were amended are reported as follows : i) The time average value of induced velocity of the propeller-race by a finite number of blades is equivalent to the values secured on the assumption of an infinite number of blades when circulation distribution is given. ii) The element-value of propeller-radius-direction in the induced velocity on the propeller disk by free vortices is not a half of the value in the ultimate wake. iii) The average value of element of propeller-revolving diredion in the induced velocity by bound vortices is equivalent to the value on propeller disk by free vortices, and the former coming over the latter, the velocity becomes zero in front of the propeller and the definite induced velocity regulated by the circulation is given in the race.