XV] OF BLOOD-VESSELS 953 



Now it is easy to shew that if, in Fig. 456, 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. Then, if 

 we make the distance DE very small, the angles Xg and x^ are nearly 

 equal, and may be so treated. And again, if DE be very small, 

 then DE'E becomes a right angle, and I2 (or DE') = I cos Xg. But 

 if L be the loss of energy per unit distance in the wide tube AB, 

 and U be the corresponding loss of energy in the narrow tube DP, 

 etc., then IL = I2L' , because, as we have assumed, the loss of energy 

 on the route DP is equal to that on the whole route DEP. Therefore 

 IL = IL cos Xg, and cos x^ = LjU . That is to say, the most favour- 

 able angle of branching will be such that the cosine of the angle is 

 equal to the ratio of the loss of energy which the blood undergoes, 

 per unit of length, in the main vessel, as compared with that which 

 it undergoes in the branch. The path of a ray of light from one 

 refractive medium to another is an analogous but much more famous 

 problem; and the analogy becomes a close one when we look upon 

 the branching artery as the special case of "grazing incidence." 



After thus dealing with the most suitable angle of branching, 

 we have still to consider the appropriate cross-section of the branches 

 compared with the main trunk, for instance in the special case where 

 a main artery bifurcates into two. That the sectional area of the two 

 branches may together equal the area of the parent trunk, it is (of 

 course) only necessary that the diameters of trunk and branch should 

 be as\/2 : 1, or (say) as 14 : 10, or (still more roughly) as 10 : 7; and 

 in the great vessels, this simple ratio comes very nearly true. We 

 have, for instance, the following measurements of the common ihac 

 arteries, into which the abdominal aorta subdivides: 



Internal diameter of abdominal arteries* 



