30 



• KNOWLEDGE 



[June 9, 1882. 



Europe, long before the time when Latin literature began — just as 

 the Knglish of our best authors is a special form of a language 

 spread over Britain and parts of Europe, long before English 

 literature began. Rome could never have forced her language on 

 so many nations. — J. V. M. Thanks. There are passages in Dr. 

 Draj>er's works which are warm, but 1 liave seen none unfair. The 

 other writer yon name (with whom, however, I thoroughly sympa- 

 thise), lias been at times a hard hitter. It is very gratifying to 

 know that you find Knowleikie free, as I have wished it to be, from 

 sly thrusts. — \V. Saixdeks. (1) For lectures on the moon nothing 

 can be more suitable than photographic slides, say from 

 Xasmyth's Moon, used either with the oxyhydrogen light or a 

 good stereoscopic lantern. (2) Probably, a submerged thicket ; 

 but careful survey would be required to determine the 

 point. — \V. S. No; the word was correctly enough printed 

 " barometer." Prof. Huxley did not, however, imply that the 

 barometer is highest on hot days, but that on the hotter of two 

 days of equal atmospheric pressure, and when therefore tbe 

 barometer should be of the same height, the mercury stands a 

 little higher in the barometer tube. This is correct, but not for the 

 reason ho assigns. It is the diminution of the specific gravity of 

 the mercury, not the change in the bore of the tube, which causes 

 the diftercnco.^A Re.\dek and ax Advkktisek. Tlianks. I quite 

 agree with you about having the articles as popular as possible. 

 I scarcely see how they can bo made much more popular without 

 becoming trivial. Of course, some, as in our mathematical column, 

 are not meant to be popular, but are for the benefit of special 

 classes of readers. But we always give a fair supply, and rather 

 more, of jxjpular matter. — J. L. Wihtaker. Wo gladly insert 

 your letter. — W. SnAW Hayleb. It was unquestionably a sun- 

 spot. Vou do not mention the power you used. — M. C. C. 

 It would not bo easy to explain a real objective change in 

 the colour of sunlight from golden to pure yellow during 

 a partial eclipse. The effect was, I should think, subjective; 

 there was a real diminution of light, and an illusory loss of colour. — 

 T. Morgan". Sorry your letter has remained so long unanswered ; 

 but the truth is, we get about three times as much correspondence 

 as we can deal with, consequently some of it is four weeks old. If 

 you send a short letter expressing your views respecting pyrological 

 matters, we will gkdly insert it. 



THE TELESCOPE. 

 A. B. C. There are only two Nebulae, 10 M. Ophiuclii and 12 

 Messier Ophinchi, in the neighbourhood of which you speak. As, 

 however, the former is very nearly on a, line joining ft Serpentis and 

 I Ophinchi, and produced as far again, while you say definitely that 

 the object you observed was above such a line, there can be but 

 little doubt that it really was 12 M. that you saw. This will be 

 referred to in due course when Ophiuchus comes to bo treated of 

 in the "Nights." Doubtless the reason why "F.R.A.S." ignored 

 a Cassiopciac as a double star was that its companion is more than 

 1' 30" from it, so that they are really two separate and distinct stars, 

 and in no legitimate sense whatever a pair or a double one. — W. 

 H. Harbis. The simple numerical formula for calculating the focus 

 of a lens equivalent to the two in a Huyghenian eye-piece is this : — 

 Divide twice the product of their foci by their sum. Let us apply 

 this to your own eye-piece, of which you say that the field lens has 

 a focal length of — ths of an inch, and the eye lens a focus of - ths 

 of an inch ; then multiplying these together (according to the 

 precept), and the product by 2, we get by 2, or — Next wo 

 54, , 12 



the object glass, by r^, the focal length of the eye-piece, we 



obtain ftj- an the magnifying power of the latter with that object- 

 glass. — J. Smith. The amplifying power of a Barlow lens depends 

 upon its position in the cone of rays from the object,-glaBS. If wo 

 call the distance of the lens from (i.e. within) the focus of the 

 object-glass d ; then if its negative focal length be 2 d, it will 

 exactly double the power of every eye-piece. In other words it 

 would give an amplification of 160 with your 80 eye-piece and of 

 800 with the 400 one. In your case this result would obviously be 

 attained if you bad a Barluw lens of 8 inches (negative) focus 

 placed 4 in. within the focus of your object-glass, or, in other words, 

 56 in. from it. 



[About three pages of "Answers " have been unavoidably held over.] 



dPur iflati;cmatiral Column. 



THE LAWS OF PROBABILITY. 



IT seems better to give the most general law for inverse probabili- 

 ties before proceeding to deal with illustrative examples. This 

 we proceed to do : — By extending the reasoning employed in the 

 last paper, the reader will have no difficulty in seeing that if there 

 are three or more bags, each containing tho same number of balls, 

 of which p in the first bag are white, q in the second, r in tho third, 

 and so on ; then, if a bag is selected at random, and a ball drawn 

 at random proves to bo white, tho chance that the first bag was 



selected i 



selected 



p + tl-t-r + &c. 



p + q- 



■&.C. 



that the second was 



the chance that the third 



and 



ill he have 



p + q + r + &c. 



any difficulty in making the requisite modifications where the 

 several bags contain different numbers of balls. Tho method to be 

 followed is precisely tho same as I employed in tho simpler case of 

 two bags, and tho result is similar, viz., that if tho chance of 

 drawing a white ball from the first bag is Ci, that of drawing a 

 white ball from the second bag C..., that of drawing a white ball 

 from tho third bag C3, and so on ; then, if a white ball is drawn 

 from a bag selected at random, tho chance that tho first bag was 



C, 

 selected is ^ — r?, — p?; — tvt ; t'le chance that tho second bag wag 



Ci-l-Cj + Cs-H&c.' 



selected : 



This result, extended to the more general case of which the bags 

 of balls are merely illustrative, becomes the following genera) 



If there are three or more hypotheses all equally likely, and 

 mutually exclusive, so that only one can be true, and if on the first 

 hypothesis the chance of an event is Cj, on the second the chanceof 

 an event is Cn, on the third C3, and so on ; then, if the event occur, 

 and we know, further, that it must have resulted from some one of 

 tho conditions inferred by the hypotheses, the ohanco that the first 



lijpothesis is the true one is -/^ — —^ — V. ,, — ; the chance that the 



second is tho true one is ■ 



Ci + C.-t-Cs-h&c. 



c; 



and so on. 



Ci + Co + C3 + &c. ' 

 One step further, and we have tho most general law of all. The 

 above law supposes all tho hypotheses to be equally likely in the 

 first instance— a state of things obviously corresponding to the 

 equal chances that any one of the bags will bo .selected. To illus- 

 trate the case where the hypotheses are not equally llkoly in tho 

 first instance, wo must assume that the chances of drawing the 

 several bags are not equal. Now, if we consider the case of two 

 bags, we shall bo able to deduce tho general law wo require. Thus, 

 suppose there are two bags, and that the chance of selecting 



; mat or seieci 



one or other must bo selected, tho sum of these proba- 

 bilities must bo equal to unity) ; and, as before, let tho chance of 

 drawing a white ball from tho first bag (if selected), be C,, that of 

 drawing a white ball from the second bag being C5. Then we may 

 represent these two chances by supposing that there are two bags 

 of tho first kind and three of the second kind, all equally likely to 

 bo taken ; for it is obvious that the chances that the selected bag 

 belongs to the f onner or latter kind arc, respectively, - and -. Now, 



by the general law already obtained— if a white ball is drawn, the 

 chance that it came from a specified bag of the first kind is, 



_-_ -.P' , or . i^i- , and tho chance that it camo 



C,-hC,-1 Co + CV-l-Cj 2C,-h3Cj 



from one or other of those two bags is ^^ +30 "- Similarly, the 



chance that it came from one or other of tho three bags of tho 



, , . , . 3Cj 

 second kmd .s ^^_^^f,^ 



Noticing how this result has been obtained, and proceeding at 

 once to the law which the bags of balls illustrate, we obtain finally 

 this general law, including all the preceding laws (of indirect pro- 

 babilities) : — 



If an event cannot happen unless some one of a set of hypo- 

 theses, III, H;, H3, Ac, be true (these hypotheses being mutually 

 exclusive), the antecedent probability of U, being r,, that of ^2,0,, 



