Let$K$be the class of trigonometric series of power type, i.e. Taylor series $$\sum\nolimits_{n = 0}^\infty {c_n z^n } $$ for$z$=$e$^{$ix$}, whose partial sums for all$x$in$E$, where$E$is a nondenumerable subset of [0, 2π), lie on a$finite$number of circles (a priori depending on$x$) in the complex plane. The main result of this paper is that for every member of the class$K$, there exist a complex number ω, |ω|=1, and two positive integers $$\nu , \kappa , \nu < \kappa $$ , such that for the coefficients$c$_{$n$}we have: $$c_{\mu + \lambda \left( {\kappa - \nu } \right)} = c_\mu \omega ^\lambda , \mu = \nu ,\nu + 1, \ldots , \kappa - 1, \lambda = 1,2,3, \ldots $$ . Thus, every member of the class$K$has (with minor modifications) a representation of the form: $$P(x)\sum\nolimits_{n = 0}^\infty {e^{iknx} ,} $$ where P($x$) is a suitable trigonometric polynomial and$k$a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of ($C$, 1) summable Taylor series.