﻿On the Hatchet Planirneter. 265 



The experiments with hydrogen can only be regarded as 

 approximately correct. 



My results for the velocity of sound in air and carbonic 

 acid in glass tubes of different diameters are in full agreement 

 with Kirchhoff's theoretical formula founded upon the con- 

 sideration of the friction and the conduction of heat of gases. 



The velocity of sound in free space for air and carbonic 

 acid is, according to my results, invariable for tones of 

 different pitch and intensity. 



It is my pleasant duty to express here to Professor Quincke 

 my heartiest thanks for his kind support and instructive 

 counsel during the prosecution of the above inquiry. 



The Physical Institute, Heidelberg, 

 January 1894. 



XXIX. The Hatchet Planirneter. By F. W. Hill*. 



^HE hatchet planirneter consists essentially of a tracing- 

 point and a convex chisel-edge rigidly connected, the 

 point and the edge being in the same plane. When the 

 point is moved along any line, the edge describes a curve of 

 pursuit. 



The object of this paper is to investigate how the instru- 

 ment may be used to determine areas. 



Let the tracer start from a point inside the area, move 

 along any line to the perimeter, then rouud the perimeter and 

 back along the same line to ; the solution of the problem 

 consists in finding an expression for the angle AOB be- 

 tween the initial and final positions OA, OB of the line 

 joining the tracing-point and chisel-edge. 



All attempts to express the area of the curve in terms of 

 this angle proved futile except in a few special cases, such as 

 the circle and square; but the magnitude of the angle may be 

 found in the form of an infinite series, the most important 

 term of which is a multiple of the area. The complexity 

 of the result would seem to show that no simple geometrical 

 explanation is possible. 



Let the tracer move a distance r along a straight line (fig. 1) ; 

 then, if %, ^' be the initial and final inclinations of the rod to 

 the line, c the length of the rod, it is easily proved that 



/ r 



tan -£- = <?" tan -^ (1) 



— - 



* Communicated by the Physical Society : read June 22, 1894. 

 Phil. Mag. S. 5. Vol. 38. No. 282. Sept. 1894. T 



