July 2, 1888.] 



KNONA^LEDGE ♦ 



201 



Let F B G be the section of a cone, having the height B H 

 very little greater than B D. Let F B, A C, intersect in a, 

 and draw F c perpendicular to A a. Then obviouslj' A a F 

 is ultimately a triangle similar to B C A. [The angle 

 ArtF=angle B«C'=iult. angle BAC; and angle «AF= 

 angle A B C] Draw C K perpendicular to A B, corre- 

 sponding, in the larger triangle, to Fc perpendicular to An 

 in the smaller. 



Now, the right-angled triangles F c « and B D « being 

 .similar, we have 



Fc : BD::ac : aT> ; 

 ■whence F c . « D =B T> . ac. 



.-. Fc. (aD)-=BD .ac.«D (1) 

 But if the cone B A C is a maximum, the increment in 

 pa.ssing to the cone B F G must be equal to the decrement : 

 wlionee idtimately (when AD^cD^^a D, etc.) 



7r.Fc(aD)-=7r 



BD 



A a. 2 « D 



(2) 



Comparing (1) and (2) we see that for a maximum we 

 must have — 



fl C=:|A« 



or AK=iAB=2KB. 



Join E A and draw D L perpendicular to A B, bi.secting A K 

 in L (beaiuse AD^DC). Then since LD and AE are 

 parallel and B L=^ A B 



'BD = § BE=i Bo; 

 the same result as by the " great engine." But the " great 

 engine " did the work more easily.* 



The Saturdatj Ileview having displaj-ed its profundity 

 thus effectively, proceeds in the following strain : — 



In tlie table of contents, we were astonished to find the heading 

 " Elliptic Integrals," [In reality the heading is " Elliptic and 

 Hyperbolic Integrals," but to have roentioned this would have 

 baulked the S. R.'s attempt at sarcasm.] but on looking up the 

 chapter found not a word about them, as ordinarily understood, 

 but only a proposition about finding the area of an fllijise by inte- 

 gration. Must we conclude that the author did not know what the 

 phrase meant ? [The S. R. naturally leaves it to be understood 

 that we must come to this distressing conclusion.] 



Now if I were given to betting, I would be prepared to 

 offer a tolen^bly large wager that the writer of this critique 

 (save the mark !) does not in the least know what an 

 " elliptic integral," as commonly understood, really is, or 

 why it is called an " elliptic integral," or for what purposes 

 the .so-called " elliptic functions " were discussed by 

 Legendre. The way in which he emphasises the word 

 "ellipse" (for the italics are his) shows this clearly enough. 

 Evidently that sarcastic emphasis would have come in quite 

 as effectively if he had had to say that he only found a 

 proposition about rectifying the arc of an ellipse by inte- 

 gration — though such a proposition would have related 

 unquestionably to " elliptic integrals " as commonly under- 

 stood. 



The fact is that the phrase " elliptic integrals," as com- 

 monly used, is a con^-enient misnomer. In the rectification 

 of an ellipse (centre as origin of co-ordinates, major axis 1, 

 and eccentricity e), we obtain the equation — 



f / i ga x^ 



s^\ A / — d X (where s represents the arc), 



and putting a;=sin 6, so that do: — cos, OdO, we get 



s=\^T. 



e de. 



* Here are two easj' problems which either the beginner in the 

 Differential Calculus or the geometrician can solve: (I) Deter- 

 mine the maximum cylinder which may be inscribed in a given 

 sphere. (2) Determine the cylinder of maximum surface which 

 may be so inscribed. 



Now the term " elliptic integrals " is applied conveniently, 

 but not correctly, to expressions of the form, 

 where ^/(x,X)dx 



X=^/ a + b.T + cx^ + dx^ + ex** 

 d and e being not both equal to zero. Such integrals can 

 be classified into others, of which all not expressible by 

 algebraic, logarithmic, or inverse trigonometrical functions, 

 have one of the three forms : — 



^'^ J Vjl-c'sir^e ^"* J ^ {l-c^-sir?ed6 



dd 



and 



(iii 



J(l+^ 



the integrals being all taken between the limitt-, .and o. 



The second form is that already obtained for the rectifica- 

 tion of the arc of an ellijise ; and for this rea.son, by no 

 means a perfect one, these integrals are all, for convenience, 

 called " elliptic integrals." 



But if mathematicians may for convenience call several 

 orders of integrals " elliptic " because one among them 

 really is elliptic, as appearing in dealing with the rectifica- 

 tion of the ellipse, they are surely free if it suits their con- 

 venience (as it did mine in the present case) to call certain 

 integrals " elliptic " which are really elliptic, as appearing 

 in dealing with the quadrature of the ellipse — especially if 

 (as 1 did) they prevent all possibility of mLstake by com- 

 bining with the term " elliptic," thus used, the term 

 " hyperbolic " similarly used, because it relates to an order 

 of integrals appearing when we are dealing with the 

 quadrature of the hyperbola. 



I wanted a convenient heading for a chapter on integrals 

 relating to the ellipse and hyperbola, so I called them 

 " elliptic and hyperbolic integrals." I knew that on the 

 one hand the readers for whom I was writing could not be 

 in any way misled by this nomenclature, especially as the 

 chapter itself sufficiently explained the use of the words. 

 I was equally well assured that no mathematician acquainted 

 with the technical (but scarcely correct) use of the phrase 

 " elliptic integrals " could be for a moment misled, if by any 

 chance he looked over my simple pages. And although the 

 idea did occur to me that critics of the S. R. class — neither 

 learners nor learned — might cavil, that did not in the lea.st 

 trouble me. I knew that on the one hand I was certain to 

 encounter cavillings of that .sort, and that on the other I 

 should know well how to treat them, and shoidd even be 

 able to apply them to a useful purpose, as I am now doing. 



That the notice I have been dealing with is feeble and 

 spiteful may be explained by the fact that it appears in the 

 paper called the Saturdai/ Review — by some (by others less 

 euphoniously). Does this remark sound unmannerly '( For 

 my own part, I think it does. I withdraw it. I was only 

 trying an experiment. The S. E. says that the defects 

 (invented) which it pretends to recognise in my two small 

 and unpretentious books, " First Steps in Geometry " and 

 "Easy Lessons in the Differential Calculus," "may be ex- 

 plained by the fact that they consist of papers written for 

 the magazine called Knowledge." I wanted to see how 

 such a remark applied even to the Saturday Revikr might 

 sound. 1 cannot thick that it sounds well. 



As for the remark thus rudely made by the S. R., it 

 chances to be untrue — no very strange chance of late, when 

 the S. R. is in question. The substance of the two books 

 dealt with was not written for Knowledge, though it ap- 

 peared in Knowledge. (The Saturday Review never could 



• If under the radical higher powers of x appear than the fourth, 

 the integrals thence deducible are called " ultra-elliptic " or " hyper- 

 elliptic." 



