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Ah, I see it’s a symmetrical dodecahedral conservatory with rich Jacobean detailing, on raised dais over heating chamber. Ashlar. Bays divided by engaged roman Doric Columns. Large window in each bay; door to S, extended from window. Moulded architraves with decorative modillion at centre. Base, lower part of columns and central chimney stack strapworked. Jacobean motifs to capitals and Greek Doric entablature. Shaped strapworked motifs over bays at parapet level. Columns originally surmounted by spiked vases. Originally small-pane glazing pattern in 3-light sash and case windows, 24-pane with 16-pane side-lights; glazing now missing, some astragals remaining and frames largely intact. Steps up to terrace from S, W and E; some newels remaining of former strapworked balustrade. Moulded panels to wall to N below terrace level; door to vaulted heating chamber and flanking windows, with bolection-moulded and shouldered architraves. The Conservatory was conceived as the centrepiece of W S Gilpin’s parterre design, little trace of which now remains. The heating system consisted of two hot-water boilers and furnaces located in the vaulted cellar; the furnaces were directly connected to the main flue within the central chimney. The strapwork is probably derived from Wendel Dietterlin’s ARCHITECTURA. (Historic Scotland)
[Another photo](http://i.imgur.com/T72bQPC.jpg), info from the rather amazing [RCAHMS](http://canmore.rcahms.gov.uk/en/site/53523/details/dalkeith+policies+conservatory/).
It’s also on the Buildings At Risk Register: http://www.buildingsatrisk.org.uk/details/906259