380 CIRCULATION 



Angle of Origin of Vessels 



One further point making for the economical working of the 

 inland transport service, owes its enunciation to John Hunter. 

 He wrote, " To keep up a circulation sufficient for the part and 

 no more, Nature has varied the angle of the origin of the arteries 

 accordingly." Suppose a point C is h units vertically distant 

 from an artery AB, the problem is to find out the route by which 

 the blood could be conveyed from A to C with the least possible 

 loss of energy. This is not necessarily by the shortest route or 

 by the route using the shortest piece of branch tubing. The 

 shortest route would be h units long and would arise from AB at 

 right angles (say at D). For the purposes of this calculation 

 let us consider that the least loss of power occurs when the branch 

 originates at X which is x units from D, making an angle of 6 

 with the main trunk. Then the distance from X to C would be 

 Vos^ + h^ (hypotenuse of right-angled triangle). 



Assuming that loss of pressure is due to friction on the walls of 

 the vessels, then it will be directly proportional to their lengths 

 and indirectly proportional to their radii {e.g. main trunk = R 

 branch = r) : 



. , . . ^ AC AX 



I.e. loss IS proportional to + -^• 



If the whole distance from A to D be put = b, then AX =^ b — x. 



Vx^ -{- h'^ b — X 

 Substitutmg, we have }- — ^ — , 



multiplying by Rr gives us the value 



S ^ R V cTa + K + {b - x)r, 



where S = loss due to friction. 



Differentiating and equating to zero we obtain a value for x 

 ^vhich makes S a minimum. 



dS 2Rx 



Thus J— — ^ ,' = — /• =^ ; 



dx 2\^x^ + h^ 



r _ X _XD _ 



R " V^^Th^ - AC - ^"' ^' 



That is, the angle of origin required is such that its cosine is 

 numerically equal to the radius of the branch divided by the radius 

 of the 7nain trunk. 



The size of the angle of origin is governed neither by the radius 



