Deo. 19, 1884.] 



♦ KNOWLEDGE 



497 



regard P M as a strip of the semi-ellipse A h A', and Q M 

 as a strip of the spmi circle A 1! A', we have 



Strip PM : strip QM :: C6 : CB, 

 aud the like with all such strips into which the areas 

 A 6 A' and ABA' may be divided by a multitude of 

 parallels to B B', indefinitely near each other. Hence, 

 since each strip in A 6 A' bears a constant ratio to the 

 corresponding strip in A B A', all the strips in A 6 A' 

 together, bear that tame ratio to all the strips in ABA' 

 together : that is to suy, 



\ ellipse Ah A.' : ! circle A B A' :: 6C : B C, 



or ellipse A J A' 6'= . circle A B A' B'. 



^ BO 



In like manner, beginning with the ratio 



P?;j : q m :: QC : qC, 



BC 

 we can prove that ellipse A b A' 6'=:y-p . circle a b a' V. 



bG 



It is also evident that area P A P' = 



BC 



area Q A Q' j 



, bC , BC , , 



area P A' P =-j7p area Q A' Q' ; area P bp = 5-p qb q ; 



BO 



and area P b' il ■=■ -r-pr area q b' q'. 



Fig. 1. 



A. I see that you have here taken the lines P P', Q Q', 

 P p', and q q' to represent elementary strips of the areas 

 you are dealing with. Can you always do this ? 



J/. You can do it in all such casfS as this, where you 

 know that the strips in one area are of the same breadth 

 as the corresponding strips in the other. Here they are 

 so by the nature of the construction itself. But in other 

 cases — in dealing with the parabola, for example — they 

 are not so. In such cjses you must consider the breadth 

 as well as the length, and therefore you cannot use the 

 lines as in the last case — for a line represents length, not 

 breadth. 



A. The ellipse seems easy enough to deal with. I 

 wonder why there should be more trouble with the hyper- 

 bola, which resembles the ellipse in so many properties. 

 How is this'? 



jy. You say the elli]3se is easy to deal with. Have 

 you satisfied yourself on this point 1 



A. Why, have we not just determined its area 1 



M. We have shown that its area bears a certain pro- 

 portion to the area of certain circles. But is the deter- 

 mination of the area of a circle so eaty a matter ? I 

 imagine that the quadrature of the circle has been con- 

 sidered a problem of some difficulty. 



A. Still it has been dealt with so thoroughly that w* 

 know the area of a circle of given radius as cloi-ely as we 

 can ever want to kn6sv it. 



J/. Quite 60. And if we had as familiar a knowledge' oif 

 the curve which biars the .'ame relation to a hyperbola thaA 

 the circle bears to the ellipse, we ."should be able to deal as 

 easily with the hyperbola as with the circle. ii.mi 



A. What curve is that ? " 



J/. What else but the rectangular hyperbola? The 

 circle is an ellipse with equal axes, the rectangular b'yp'eir- 

 bola is a hyperbola with equal axes. 



A. Then, in point of fact, all we have done with ths 

 ellipse has been to show what relation the quadrature 61 

 the ellipse bears to the quadrature of the circle. 



J/. Precisely. And it is just as easy, as a moment's cor> 

 sideration will show you, to see what relation the quae)' 

 rature of a hyj)erbola with unequal axes bears to th» 

 quadrature of a rectangular hyperbola. "' f'"' ' 



A. Will you take a momen;'s consideration for Jne, and 

 show me this ! 



J/. Suppose in Fig '1, C A and C B are the .', axes of a 

 rectangular hyperbola AQK; OA and 6 those of tW 

 non-rectangular hjperbola AP^ ; ODE F, C de f, asyiajv 

 totes; DA, EQPN, FK/.M perp. to CM. Theo we 

 know that 



PN:QN::/.:M:KiM::CB:C6. 

 Hence, as in the case of the ellipse. 



Area A AM: area AKM::CB:C6; 

 and if we had dealt as freely with such areas as A K M^ 

 (that is the areas of the segments of rectangular hyper- 

 bolas) as we have with the areas of circular segments, this, 

 would solve as completely the problem of hyperbolis 

 quadrature as the preceding investigation solved ths 

 problem of elliptic quadrature. 



A. That is far from satisfying. Allow me to say witii 

 Shakespeare, " Out hyperbolical fiend." Let us turn to the 

 parabola. 



(To he continued.) 



THE ENTOMOLOGY OF A POND, 



By E. a. Butlee. 

 ABOVE THE SURFACE {continued). 



IN a former paper a small family of moths was referretJ 

 to as having aquatic caterpillars ; these will form th>^ 

 subject of the present paper. They aie called the H}dro- 

 campidse, or China Marks, and are common adornments of 

 weedy ponds. They are but a feeble baud, the British- 

 species being only five in number, distributed amongst 

 almost as many genera. The largest is but an inch in 

 expanse of wings, aud the smallest little more than half 

 that size. The wings are rather narrow, and often mosi 



