264 



KNOWLEDGE. 



July, 1911. 



two hundred and t\vent\'-t\\o. and since tliis in turn 

 produces radium A and radium B successiveh', with 

 the loss of an a particle in each case, the atomic 

 weight of radium B should be two hundred and four- 

 teen. The method of recoil thus gi\(-'s an important 

 conhrmation of this theoretical deduction. 



An account has been given of some new branches 

 in radio-activity w hich have already been opened to 

 investigation by the method of recoil. It seems not 

 unlikel}- that man\' hitherto unsolved problems 

 may recei\-e explanation when attacked b\- this 

 recentlv-discovered method. 



OX DRAWING ELLll^SES. 



liv \V. B. GIBBS. F.K.A.S. 



In the numlier of the Joiinuil of the British 

 A.stniiioiniLiil Association, for .Ma\', 1910. Mr. 

 Hardcastle has given an interesting accoimt of 

 a method of forming the boundaries of an elliptic 

 area bv folding down segments of a paper circle 

 according to the following rules : — 



The paper circle is to be about six inches in 

 diameter, the centre S is to be marked and also 

 some other point H, which it is convenient to 

 choose about two inches from the centre. On the 

 circumference a number of points are to be marked. 

 I myself find about thirty are desirable. K (Figure 1). 

 will serve as an example of one of these points. The 

 paper is then to be folded so as to bring K to H. and 

 the crease is to be marked clearly. Then the paper 

 is to be opened and the next jioint on the circum- 

 ference is to be brought to H. ami tiiis new crease to 

 be j^ressed clearly, and so on for the whole number 

 of points. Then it will be found that a neat smooth 

 area is left forming an ellipse al)out .S and H, each 

 crease being a tangent to the curve. 



Now I find that this elliptical area can be made 

 much more distinct, if. when the segments are folded 

 down, thev be shaded on the outer side with a lead 

 pencil and then this shading well rubbed with the 

 finger. Figure 2 shows an example of a shaded 

 segment. When all the segments have been so 

 treated and the paper is turned up, there will be seen 

 on the lower side a clear elliptic area as shown in 

 Figure 3, the tangent crease lines being also ver\- 

 plain. 



Mr. Hardcastle"s paper is well worth reading in 

 its entiret}-. He gives a geometrical proof 'that the 

 area is bounded by an elliptic curve, and also shows 

 that it is not only interesting geometrically but that 

 it can be used for dynamical purposes. 



It is easily seen that ellipses of any degree of 

 eccentricity can be drawn in this way. for from 

 the properties of the ellipse .\S = .\'H = .\'K there- 

 fore .■\.\ is always equal to SK. or the major axis of 

 the ellipse is equal to the radius we have chosen for 

 our paper circle, while the minor axis continuall\- 

 decreases as H approaches K, vanishing absolutelv 

 when H coincides with K, the ellipse then becoming 

 a straight line : but if H approaches S then the minor 

 axis gradually increases, and ultimatel}', when H 

 coincides with S, becomes equal to the major axis, 



and the ellipse becomes a circle whose radius is equal 

 to one-half the radius of the original paper circle. 

 If the radius of the original be 3 inches 

 Then AS = J -inch 

 and the minor axis =^ l|-inches 

 from formulae gi\en in all treatises on conic sections. 

 In order to draw elliptic orbits b\- this method, I 

 subjoin the accompan\'ing table giving the eccen- 

 tricities of the orbits w hich correspond to those values 

 of SH which lie between two and three inches, 

 proceedmg b\" thirt\'-seconds of inches. This will 

 be available for all comets which mo\e in closed 

 orbits. Taking, for example, Encke's comet, the 

 eccentricit\' of which is •8476, we see at once we 

 must take SH between 2g| inches and 2^\^ inches, 

 because -8476 lies between -84,^75 and -8541, the 

 nearest numbers given in the table. 



