106 PROFESSOR A. H. GIBSON ON THE RESISTANCE TO FLOW OF WATER 
greater when 6=20'—-than in the corresponding circular pipe, while for values of @ 
ereater than 8° it is also greater than in a rectangular pipe with the same value of 0 
but having one pair of sides parallel. 
§ 6. Pipes or Best Form. 
Where no restrictions are placed on the length of the pipe, a straight taper pipe 
having a divergent angle of about 6° in the case of a circular pipe and of a square pipe, 
and of about 11° in the case of a rectangular pipe, will give the minimum loss of energy 
between inlet and outlet.* The great length of such a pipe, however, renders its use 
impossible in many cases which occur in practice, and in such a case it becomes im- 
portant to determine what form of passage will give the least loss for a given length 
and given ratio of enlargement. It would appear that such a pipe should be trumpet- 
shaped, the angle of divergence being least at the small end of the pipe where the 
velocity is greatest, and gradually increasing as the velocity diminishes. 
While impossible to determine the best curve from purely a@ przori reasoning, it 
seemed reasonable to suppose that the loss might be least either 
(1) with a pipe giving uniform retardation (“7=constant) ; 
or ; 
(2) 5 As , change of velocity per unit length of pipe (2 = constant) ; 
a 
or 
(3) * - <s loss of head per unit length of pipe. 
In the former experiments pipes were prepared both of circular and of rectangular 
section for the purpose of testing the validity of the first two of these assumptions. t 
In the case of the rectangular pipes, which had the same initial and final areas and the 
same length as a straight taper pipe having 9=22° 42’, it appeared that the loss was 
reduced by 5°3 per cent. in the former case (7) = constant) and by 12°1 per cent. in 
d : 
the latter case (5 = constant). In the case of the circular pipe, however, the loss was 
actually greater in the pipe having . constant than in the straight taper pipe. On 
76 
this account it was decided to test the validity of the third of these assumptions, and 
the pipes referred to in the table opposite were prepared for comparison with straight 
taper pipes of the same length and same change of section. 
The boundary curves for these pipes were set out from equations deduced as follows:— 
The loss in a straight taper pipe whose angle of divergence is @ is proportional to 6(v)? 
ayn 
and very sensibly to @” or to (tan 5) where n= 1°40 for a rectangular pipe. Hence in 
* This statement requires modification in view of considerations outlined at a later stage of the paper. 
+ Ibid., pp. 367, 375, 376. 
; Not 20° as stated in the former paper. 
