In 1997, the author considered the separate coding system for two correlated memoryless Gaussian sources and squared distortion measures and determined the rate distortion region in a case where one source plays a role of the partial side information at the decoder. The above source coding system can be extended to a certain class of source network with several decoders, where each decoder has at most one full or partial side information. This class of source network is called the one-helps-one system. In this paper we consider a source network belonging to this class for correlated memoryless Gaussian sources and squared distortion measures. This source network was posed and investigated by Körner and Marton and was called the zig-zag source network. They studied the zig-zag source network in the case of discrete memoryless multiple sources. In this paper we study the zig-zag source network in the case of correlated memoryless Gaussian sources and square distortion. We determine the rate distortion region in a case where sources have a certain correlation property.