Oct. 6, 18S2. 



KNOWLEDGE ♦ 



31 



at B varies inversely as the square of the distance B L, and directly 

 as tl,e sine of the an-le LliA. Hence, taking m (wliicli will „o't 

 eventually tronblo us) as a suitable constant, we may rei.rcsent the 

 ilhunination at B by the expression 



m AL m.AL 

 ^ „ , (bl>'b~l= BL» 



Call this illumination y; then 



"^ <■■' 



Now, to differentiate this expression, we take first the differential 

 coefficient of the nnmorator and multiply it into the denominator. 

 Ihis gives m(l + x')i 



Then we take negatively the differential coefficient of the deno- 

 minator and multiply it into the numerator. This gives 



-^(l + .')t.2^xm.r,or3(l + .r')lma.= 



We have only to put the square of the denominator under these 

 two expressions, thus 



(1- 



^) = -.3(l. 



=)»»,,: 



(1 + ^')' 

 to get the differential coefficient of y, and we could reduce and 

 simplify this expression considerablv. But observe, we onlv want 

 to equate the coefficient to zero (that being, as in former cases, the 

 way of getting the required value of x). . So that we have no 

 occasion to write out the differential coefficient as above, or to 

 reduce It We need only write its numerator and equate that to 

 zero. ^^"^S^y^^ ni(l + x')i-3il + x^)imx''^0 (iii.) 



or, (1 + .c-') = 3.V- [dividing out bv m (1 + x')!] 



That is 2x^=1 



.r=-L 

 V2 

 Hence A L should be about 8^ in. 



If we had taken A B = a instead of 

 the result : — u.-.,..., a'' + x^ = 3j 



<■"■, 2x^ = a'-' 



should have obtained 



v'2 



Note that tlie writing of this problem would be very phort in 

 practice. Jn fact, after expressing y in terms of x, as at (i.), the 

 student would write down equation (ii.) at once; and solve as 

 above. 



Observe, again, that wo can vary onr method of attacking these 

 problems by varying the relation whose change is to give our 

 iii.'ixmmm. For instance, suppose we attacked the above problem 

 in this vrise : — 



l.rt the angle L B A, Fig. 2, be called a-. Then we know that 

 E L = B A sec x. And the illumination at B may bo represented by 



B A' sec 

 Putting this equal to ;/, we have, 



Hence 



[T le, 



■.m[-2 



' X cos a] 



vo the reader to see how this is obtained by applying the 

 x-uies given in preceding papers, noting that a very niodenito 

 degree of practice will enable him to write down such a result at 

 once.] 



Kquating this expression to zero, we have 



cos' X cos r = 2 cos it sin' x. 



Hence £1IL£^1 



cos' » 2 

 .._ _ 1 



v/2 



A L 

 H a' 



"('"CO A L = --i^- = — as before 



Neither method has the advantage in this instance, but often a 

 good deal depends on a proper choice of the method to he fiillowi-d. 

 For example, suppose B is a point on a desk (Fig. 2) sloped at an 

 anprle ri to the horizon, and that wo require the height A f, which 

 will f,'ivi' the inaximum illumination in this case. Then the first 

 metlHid would be very inconvenient ; but, on applying the seeond, 



""-' ^■»' 'f V i« little altered. We onlv have to p'lit, instead of the 



factor sin x (i.e., sin LB A), the value sin (x-c) [i..:, sin 1, It M ', 

 Pivn'g y = mco8'a:8in (.r-a). 



And this is differentiated quite a? easily as the other expression, 



giving i- . ^' = -2 cos ar .sin x sin (.--«) 



J!i dx 



+ era' X CCS (x — a). 

 Equating thi^ to zero, we 1 ave, 



2 cos X sin x sin (x—a) = 

 2sina^sin(;r-a)=cc 



cos (x — a) 

 {x-a). 



(iii.) 



Here we want only a moderate familiarity with trigonometrical 

 processes to get out our result, for equation "(iii.) is the same as 



2 cos a -2 CCS (2,r— a)=cos a + cos (2x—a), 

 that is 3 cos (2x-a) =cos o, 



This is sufficient for finding x, because a is supposed to be known. 

 Suppose, for example, a is equal to eight degrees, the slope of my 

 desk, for I have taken the notion of wnrkivg out this particular 

 problem u'iih the practical desi.jn of determining how hijh I should 

 sft the moderator u-hich illuminates the paper I am uritinij upon ; 

 wo have then, 



coi (2a! -8") = J cos 8° 



= i (•99027) from a table of natural cosines 



- -33009 



= cos (70° 43") nearly enough 

 .-. 2j;-8' = 70"44' 



And A L, Fig. 2, is equal to B A tan t. Now, in the case of my 

 desk and light, B A is equal to 18 in., and B, the part of the desk 

 where I actually write (shifting the paper to this point as I write 

 on \\n? by line) is about 4 in. above level of table, so that the height 

 of the light should be (4 + 18 tan 39° 22') in. 



I take out the log tan of 39° 22', which is 1 91401, and add to it 

 the logarithm of 18, which is 1-26527, getting 1-1G931, which is the 

 logarithm of 14-7(i8. Adding 4 to this, I find that the best height 

 for the light of the moderator (above surface of table) is as nearly 



as possible IS] in.* 



lU.ssuNs employ petroleum successfully upon fome of their 

 railways for driving locomotives, using for this purpo.se the crude 

 naphtha as it comes from the wells. Most of the steamers that ply 

 the Caspian Sea use the liquid fuel, wliich is very much cheaper 

 than coal. It is consumed with injectors fuch as are used in this 

 country, and the combustion is regulated with the greatest case. 



• The reader well u]) in his " Gulliver" may be reminded hero 

 of the tailors of l.nputn, who apjili. d matliematical methods in 

 taking their customers' measure, and were led semelinics by errors 

 in calculation to make ill-fitting garments. An error in the working 

 (if the above problem might easily (for instance) have given l-87in., 

 or 187 in., instead of ISJin. ; and the care rccessnry to secure a 

 tine result might well seem greater than the probtim, practically 

 viewed, is worth. I have indeed merely taken the pn blem as a 

 convenient illustration of the fact that the dimrenlinl cahnhis 

 may often bo a|>plied to practical matters, without dcsii ii g to 

 insist on the special value of the result tbtainid in this particular 

 instance; though, be it noted, that as the light of my lamp stood 

 but IGi in. high, I set the lamp 2} in. higher after I had woikod out 

 the above result. 



