We may follow Hess in a further inôÙvesôÙtiôÙgaôÙtion of this phenomenon. Let AB be an artery, from which a branch has to be given off so as to reach P, and let ACP, ADP, etc., be alternative courses which the branch may follow: CD, DE, etc., in the diagram, being equal distances (=ã₤l) along AB. Let us call the angles PCD, PCE, xÿ£¢1ã₤, xÿ£¢2ã₤, etc.: and the distances CDÿ£¢ãý, DEÿ£¢ãý, by which each branch exceeds the next in length, we shall call lÿ£¢1ã₤, lÿ£¢2ã₤, etc. Now it is evident that, of the courses shewn, ACP is the shortest which the blood can take, but it is also that by which its transit through the narrow branch is the longest. We may reduce its transit through the narrow branch more and more, till we come to CGP, or rather to a point where the branch comes off at right angles to the main stem; but in so doing we very considerably increase the whole distance travelled. We may take it that there will be some intermediate point which will strike the balance of advantage.
Now it is easy to shew that if, in Fig. 330, the route ADP and AEP (two contiguous routes) be equally favourable, then any other route on either side of these, such as ACP or AFP, must be less favourable than either. Let ADP and AEP, then, be equally favourable; that is to say, let the loss of energy which the blood suffers in its passage along these two routes be equal. {669} Then, if we make the distance DE very small, the angles xÿ£¢2 and xÿ£¢3 are nearly equal, and may be so treated. And again, if DE be very small, then DEÿ£¢ãýE becomes a right angle, and lÿ£¢2 (or DEÿ£¢ãý) =ã₤lã₤cosã₤xÿ£¢2ã₤.