ON A NEW INSTRUMENT FOR DRAWING ENVELOPES. 619 



A Neiv Instrument for Drawing Envelopes, and its Application to the 

 Teeth of Wheels and for other Purposes. By Professor H. S. 

 Hele-Shaw, LL.B., MJnst.G.E. 



[Ordered by the General Committee to be printed in extenso.'l 



The present paper is divided into two parts, the first dealing with a 

 description of the various forms of the new instrument, the second dealing 

 with the various examples of its use as a trammel for drawing curves, and 

 of its application for various purposes, such as the teeth of wheels, blowers, 

 and other revolving bodies where moi-e or less accurate contact is required, 

 necessitating the formation of an envelope to a curved outline previously 

 described. 



A communication which may be regarded as supplementing the present 

 paper has been read in the Mathematical Department of Section A (see 

 p. 136), dealing with the applications of the instrument for the con- 

 struction of cycloids and involutes, and for the drawing of ellipses and 

 other curves. 



Some time ago the author devised an arrangement for exhibiting to 

 his students the envelope of any plane figure revolving about a fixed axis 

 upon another revolving surface, the two rolling together upon a pair of 

 imaginary pitch circles. This arrangement merely consisted of two sheets 

 of papei' turning on drawing pins as centres, while a small wheel, with a 

 number of projecting needle points on its edge, simultaneously engaged 

 both sheets of paper, thereby compelling them to revolve together. This 

 wheel of course was really in contact with the two imaginary pitch circles. 



It was then seen that it was not necessary to employ the actual pitch 

 circles, as auxiliary circles would serve the same purpose and would not 

 necessitate the paper or cardboard being actually punctured by the small 

 needle points. Moreover, the circles were formed separately on each sheet 

 so long as the respective wheels were connected by one axle, and narrow- 

 milled rollers could be used instead of the roller with needle points. 

 What was more important, the space left when parts of the surface cutting 

 the pitch circles, such, for instance, as the spaces in tooth gearing were 

 actually removed, did not prevent the continuous rotation of the two 

 sheets of paper. A pencil outline round the boundary thus left could be 

 drawn in a number of positions close to each other, thereby giving the 

 required envelojje for the corresponding teeth required to gear with those 

 originally formed. 



Fig. 1 shows the instrument, which had two pairs of roller or disc 

 wheels (connected by one axle), one above and one below the moving 

 surfaces of the cardboard. 



With this instrument very little slip took place, and it was found that 

 students could easily use it without having had any previous experience. 



There are three ways in which a given curved outline of any form 

 working about a fixed centre could be obtained. 



(1) By stencilling in the outline of the revolving sheet beneath. 



(2) By photographs, when the two surfaces moved over one another 

 with a fairly rapid movement. 



