Formula for the Discharge over a Broad-crested Weir. 95 



rotation, i. e. the disk has become a straight line coinciding 

 with the axis of rotation and of length equal to the original 

 radius of the disk. 



This straight line, of infinite density, is analogous to the 

 plane of infinite density obtained by moving a solid body in 

 a straight line with the velocity of light. 



The difficulty of experimentally discriminating between 

 this solution of the problem and the solution which considers 

 the rotating disk as contracting but still remaining in one 

 plane would be great. If we assume that light is reflected 

 from a mirror fixed normally to the disk, and assume that we 

 can detect a deflexion of 1 mm. in the position of the reflected 

 beam received on a scale 10 metres distant, i. e. a value of <j> 

 equal to 20W0' ** w ^ s ^ ^e necessai T to have a frequency 

 of revolution of about 1000 per second to produce this effect. 



In conclusion, it would seem probable that, for a disk of 

 any appreciable thickness, the plane position would be main- 

 tained during the rotation, the material of the disk being- 

 strained, in which case Ehrenfest's contention, that we have 

 here a contraction of a line in a direction perpendicular to 

 its direction of motion, is valid. On the other hand the 

 above theory does away with this difficulty, but involves a 

 change of form of the disk, which does not, however, lead 

 to any conclusions not in perfect accordance with relativity 

 principles. 



A ! 



VII. A Rational Formula for the Discharge over a Broad- 

 crested Weir. By Professor A. H. Gibson, D.Sc, 

 University College, Dundee*. 



S was first pointed out by Dr. W. C. Unwin, an 

 expression for the flow over a broad-crested weir may 

 be deduced from first principles if it be assumed that the 

 crest is so wide in the direction of flow, that the water 

 settles down before leaving the crest, to form a parallel 

 stream of thickness t, and that in this stream the pressure at 

 any point is that corresponding to its depth. Thus, assuming 

 the velocity in the surface, and at every point in this stream 

 to be given by \/2g()l — t) ft. per second, where H is the 

 up-stream head measured above the crest, the discharge is 

 given by Q = bt s/lg(\X — t) cub. ft. per second, where b is 

 the breadth of the stream. 



As the stream will adjust itself so as to give maximum 

 discharge under given conditions, t can be determined by 

 * Communicated bv the Author. 



