September 1, 1888.] 



KNO\VLEDGE 



259 



The geometrical solution in this case is easy enough, as it 

 chances. In the following case the geometrical solution 

 is decidedly inferior to the analytical. For the area of the 

 cylinder, equating the increment to the decrement, -we 

 Lave — • 



2x2r . OL. wp=2 . 2 7r. 2 (PL. OL-;W.OZ), 

 or 0L.n;>=2 [PL . OL-(PL-nP) (OL + Ll)]; 

 = 2 (jiP . O L— P L . np) ; neglecting n P . L I, as 

 ultimately evanescent compared with the other terms in the 

 equation. 



FL_ 2nP— ?tp _ 2 0L— PL 



■■ OL 2np -ItL 



(by similar triangles P^jm, OPL) or •2{0L'^-V1J) = 

 OL. PL; 



whence, 2 (^^J j^-^ [;) =1, 



i.e., 2cotPOL-2tanPOL=.l ; 

 whence 2 cos'^ P L-2 sin^ P O L=:cos P L . sin P L, 



the same rel.ation as by the other method, since PO L=^, 

 but obtained in a far less satisfactory manner. 



The student will find it a useful exercise to deal next, 

 first by the differential calculus, and next by geometry, with 

 the following problems : — 



Determine the cone of maximum area (i.) loithout inchulinri 

 base, and (ii.) including base, which can be enclosed loithin 

 a given sphere. 



A correspondent in Bombay calls my attention to some 

 supposed misprints in my " Easy Lessons in the Differential 

 Calculus." One or two of these cases ai'e real misprints, 

 too easily to be detected, however, to requii'e correcting, the 

 rest are mistakes on my correspondent's part. He asks 

 further this question: — It is stated on p. 14 that if we 

 know the rate of increase of a quantity s (as another, t, 

 varies) to be a certain quantity gt, and also know this quantity 



gt to be the differential coefiicient of another quantity i— , 



with respect to t, we can at once write 



s=(7«- + some constant. 

 How, asks my correspondent, can this be regarded as 

 obvious 1 The answer is that the differential coefficient of a 

 quantity with respect to another has been already defined to 

 be the rate of increase of the former as the latter varies. 

 So that such a statement as my correspondent finds difficult 

 is in reality akin to such a statement as the following : — If 

 we know that the rate of increase of a certain property per 

 annum is 250Z., and also know that the only portion of the 

 property invested at 5 per cent, shows this annual increase 

 to be the interest on .5,000?., we can at once write 

 The property=5,000Z. + a constant portion bringing in no 

 interest. 



6 £J£JIP» 



By Eichaed A. Proctor. 



Mb. T. Fr.\ser is very anxious that I should deal with 

 certain difficulties which have occurred to him in reading 

 my " Old and New Astronomy ; " and he is apparently 

 not aware how fully the consideration of these difficulties 

 will show the unlikelihood of his being able to deal with 

 the overwhelmingly difficult problem of the cause of gravity, 

 which he has made the subject of a pamphlet. I will 

 endeavour to discuss matters for him. 



First, he asks me how I can reconcile my statement in 

 " Old and New Astronomy " that were the eccentricity of 

 the earth's orbit to disappear she would move in a circle 



having a diameter equal to the major axis of the elliptic 

 orbit, with my statement in Knowledge that " if the 

 direction of a planet's motion when the planet is at its 

 mean distance is right, it will travel in a circle having that 

 mean distance for radius." Considering that the mean 

 distance in an elliptic orbit is equal to half the major axis, 

 while the radius of a circle is half the diameter, the two 

 statements seem to me to be rather obviously identical. 



In sec. 502, p. 215, the words ''fig. 139" are, .as Mn 

 Fraser points out, a misprint, rather obviously for " fig. Hi." 

 If an author, in his anxiety to be understood, adds new 

 figures to make his original explanation more complete, it 

 will occasionally happen that all the references to the figures 

 whose numbers get altered will not be corrected. Still, no 

 reader who is really interested can, in such cases, be for a 

 moment misled ; and every such reader will prefer full 

 illustration to mere precision of references to illustrations 

 manifestly insufficient. As the number of figures in "Old 

 and New Astronomy," including plates, will run to over 

 six hundred, the ingenuous and generously minded reader 

 will perceive that there is more room for an occasional slip 

 (not really disturbing the student's progress for a moment) 

 than in works where the illustrations are counted only by 

 tens or twenties. Mr. Fraser's supposition that H H' in 

 section 503 should be h h' is mistaken. 



* * * 



Mr. Fraser objects to the use of the same letters, 

 accented or unaccented, capital or small, in such cases, as 

 they involve a strain on the memory, because you have to 

 siy " capital H," " small h," " accented H," " unaccented A," 

 and so on — whereas he does " not know any argument by 

 which the practice can be defended." " Excuse me, if I 

 seem to lecture you," he proceeds (I very readily do), '• but 

 this has been on my mind for some time, and I feel better 

 for the removal of the load." I apprehend he ought rather 

 to say, "Do not smile if I seem to lecture you," for he is 

 writing of a matter about which, as he practically admits, 

 he knows nothing ; whereas I have had occasion to study it 

 carefully. The argument by which thepractice is defended 

 is that these repeated letters always represent associated 

 things. Thus the mathematician calls the two ends of the 

 major axis of an ellipse A and a, or A and A' ; the two 

 ends of the minor axis B and b, or B and B' ; the two foci 

 S and S' (if he sometimes writes II for one focus it is 

 because, while S is the first letter for the sun, H is the first 

 letter of Helios, the Greek name for the sun). No mathe- 

 matician, I imagine, ever thinks of saying " capital H " and 

 " small /ij" at least I never trouble myself to make such 

 changes in reading any one else's mathematiail demonstra- 

 tions ; neither do I say " non-accented H " and " accented 

 11," but simply "H" and "H dash." However, if Mr. 

 Fraser's mind feels free of a load, what more can be 

 desired ? 



" But now," continues Mr. Fraser, " I come to a matter 

 which has been on my mind for a very long time, in fact 

 ever since 1881." I promised to deal with the tides, and 

 Mr. Fraser " is sorry to tell " me " that after reading a part," 

 and a part only, of what I say in Part IV. of " Old and New 

 Asti'onomy," he is "in as great a fog as ever." He " happens 

 to have invented the centrifugal theory for himself," and so 

 takes " a fatherly interest in it." Seven years ago he thought 

 himself the first inventor of it. And " I always thought it 

 an extraordinary coincidence," he proceeds, " that iu the 

 same number of Knowledge in which I expected some 

 notice of it to be taken, appeared a review of Professor 



