96 P. Dutt — Properties of the circle and analogous matters. [No. 2 



WehaveiP = PPxg Fig ' 5 ' 



»i> AD DE 



= BP cot 2 DAE 



= a cot 2 DAE. 



Therefore, the position of the point 



P is known if the magnitude of the 



angle BAE is given. 



Also for any point on the circle described round B with a fixed 



radius AP tan 2 BAE is constant and equal to the radius. (17). 



AP 

 Now, if .AP, BP vary but the ratio -j-- remains constant, the ratio 



AD 



■ — will remain constant, and also the angle BAE. 

 BD 



Therefore, if A be fixed and any point B is taken in AB, and D 



AD 



taken such that -— is constant, the locus of the point where the per- 

 BD 



pendicular at D meets the circle described on AB as diameter is a fixed 



straight line through A. (18). 



The preceding propositions will, I hope, show the utility of the 

 method. Many other properties may be similarly deduced, and this 

 paper is put forward in the hope that it will lead to the deduction 

 of other important and useful results by others. 



I beg to add that the properties alluded to before, may, I think, be 

 utilised in some physical experiments in which two motions are produced 

 simultaneously at two different places, but as I have not the opportunity 

 or ability to carry them out myself, I only beg to give a brief outline 

 of the method in the hope that others will be able to work it out. 



Experimental measurement of the velocity of sound from observations 

 in a railway train. — If a railway train travels at a constant speed, and 

 a sound be produced at a distance at the same moment, when the 

 train leaves the station, the sound will be heard in the train at only 

 two points in the line of motion, formed by its intersection with the 

 circle with the source of sound and the station as the fixed points. 

 From the observations the ratio of the velocity of sound to that of 

 the railway train can be calculated, as shewn below. The following 

 proof applies to the case in which the line of motion is at right-angles 

 to the line joining the railway station to the source of sound. 



Let B be the station and A the source of sound in the figure in 

 Ex. 3, Hall and Stevens page 361, and let H and K be the points where 



