a time on site for all visitors trend report by the default provider  

This method can be accessed through the Analytics.asmx file located in:

~/Workarea/webservices/WebServiceAPI/Content/Analytics.asmx

[SoapHeader("AuthHeader")]
[SoapHeader("RequestInfoHeader")]
[WebMethod(MessageName = "GetTimeOnSiteTrendReport", Description = "a time on site for all visitors trend report by the default provider")]
public TrendReportData GetTimeOnSiteTrend(DateTime startDate, DateTime endDate, AnalyticsCriteria criteria);

<SoapHeader("AuthHeader")>
<SoapHeader("RequestInfoHeader")>
<WebMethod(MessageName = "GetTimeOnSiteTrendReport", Description = "a time on site for all visitors trend report by the default provider")>
Public Function GetTimeOnSiteTrend(startDate As DateTime, endDate As DateTime, criteria As AnalyticsCriteria) As TrendReportData