I was raised in the area between Dalhousie Terrace and Braid Road, so know the area well. To head for 28 Dalhousie Terrace by bus from the city centre, the nearest bus stop is in Comiston Road, almost opposite Craiglea Drive. It was there (and still is) by the time I was first around in the 1960s, and I suspect was there from much earlier - probably from when the trams came to the area in the late 19th century. However, there is nothing to say that she had arrived by bus that evening.
If you are prepared to speculate, perhaps she had been visiting somebody further up the hill on Braid Road, by the Braid Hills Hotel, or a bit southwards - many of the houses there now were built well before 1930. On her way homewards, she could have been "caught short" and decided to head through the gateway of the Hermitage of Braid to go behind a tree for a "comfort break". In the dark away from Braid Road, she may have been unlucky and chosen a tree on the downslope to the burn to go behind, and lost her footing, and fallen into the burn and drowned.
The Google Maps vehicle has been along the road into the Hermitage of Braid, so you can see the bridge over the burn, and the road running parallel to the burn as it heads eastwards.
https://www.google.co.uk/maps/@55.9193222,-3.2101293,3a,75y,143.59h,109.55t/data=!3m6!1e1!3m4!1so0tDxNEpRDUyU1t49oTN9Q!2e0!7i13312!8i6656?hl=enThe burn is less than a foot deep and free flowing in this area. At just below freezing point, the burn wouldn't be frozen over, but the muddy slope down to the burn could easily have been slippery. The surface of the burn is about three to four feet lower than road level near to the bridge just inside the Hermitage gates.
Food for thought,
ADP
