reviving old thread. if you fly sparge how/when do you take your first runnings gravity? How does this work in the equations by not having first runnings volume?
The equations in the OP don't depend in any way on sparge method, nor do they require any volume measurement other than the strike volume. The defining equation for °Plato (°P) is:
°P = 100°P * Wt_of_Extract_in_Wort / (Wt_of_Extract_in_Wort + Wt_of_Water_in_Wort)
Since Wt_of_Extract_in_Wort + Wt_of_Water_in_Wort) = Total_Wt_of_Wort, °P is just the weight percent of extract in the wort.
The weight of the water in the wort is just the weight of the strike water used in the mash. The equation in the OP assumes you are working in metric (volume in liters, and weight in kilograms.) Since 1 L of water weighs about 1 kg, volume of water in L is approximately equal to the weight of water in kg. The equation in the OP substitutes liters of strike water for kg of strike water. However, water only has a density of 1.000 kg/L at 4°C (39.2°F.) At strike temp (about 160°F) water has a density of 0.977 kg/L. So, for better accuracy, the water density at the volume measurement temp should be used.
If you are using gallons and pounds instead of liters and kilograms, the equations get a little more complicated. One gallon of water weighs 8.3304 lb at 68°F and 8.1545 lb at 160°F. The °P equation becomes the following when using lb and gal:
°P = 100°P * Extract Wt [lb] / (Extract Wt [lb] + Strike Vol [gal] * Water Density [lb/gal]), with the density at volume measurement temp being used
So, to finally answer your question: the first runnings gravity should be taken after you have assured that the wort in the mash has been well homogenized (the wort has equal extract concentration and SG everywhere), and before any sparge water has mixed with the runnings.
Brew on