July 7. 18R2.1 



• KNOWLEDGE • 



101 



contiguous particles to the ignition point, the burning cannot 

 continue without the continued application of eitenial heat. 

 The combustion of ammonia consists not only of combina- 

 tion, but also of separation, of atoms. The chemical change is 

 represented by 4NU3 + 30; = 2Xj + 60^0, the hydrogen combines 

 with oxygen, and nitrogen is liberated. The heat prodnceil by the 

 combustion of the hydrogen is, in great part, used up in dissociating 

 that element from the nitrogen. For this reason the ignition point 

 is high, and the sensible heat of combustion low.^W. P. Provided 

 the combustion of coal gas is complete, precisely the same amount of 

 heat is generated, whether the gas be burned from a Bunsen, or an 

 ordinary luminous gas-burner. In the latter case, however, the 

 combustion rarely is complete, as witness our smoky ceilings. 

 Where the flame is used for lieating, say a kettle of water, the 

 iinnscn is more efficient, because it deposits no soot ; any 

 Huch deposit acts as a non-conductor of heat, and further 

 represents fuel that ought to have been burned up. Again, 

 a luminous flame radiates heat as well as light, and this is 

 another source of loss. In gas cooking-stoves, meat is roasted by 

 the radiant heat, and here small luminous jets are always employed 

 above and around the joint, but not under. The advantage of the 

 Bunsen under certain circumstances depends, not on its posses.sing 

 a power of producing absolutely more heat, but on its producing it 

 in a form more .ipplicable to the purpose on hand. If instances are 

 wanted of the densest ignorance of the elementary laws of heating, 

 glaring examples can usually be found in any exhibition of cooking 

 and heating-stoves. — Cekto. " Dnlong and Petit, who were the 

 earliest investigators on the subject, contended that all elementary 

 atoms hare the same capacity for heat, or, in other words, that 

 the specific heats of all elementary atoms are the same. If this 

 law be admitted, it is obvious that the determination of the 

 specific heat of an element must furnish a ready means of 

 fixing its atomic weight." (Watts's Dictionary.) This explains 

 Frankland's rule ; since 7 is the atomic weight of lithium, " the 

 weight of any other element in the solid condition which at the 

 same temperature contains the same amount of heat, as 7 parts of 

 lithium must be the atomic weight." This rule is particularly useful 

 in determining whether a particular number or a simple multiple of 

 it really represents the weight of the atom. In the case of solids 

 it is often difl[icult to determine this otherivise. Lithium is con- 

 veniently selected as the st.andard because its atomic weight is low. 



The rule, with ecjual propriety, might read, "the weight 



as 2-i parts of solid magnesium." Lithium has a specific heat of 

 O04O8. Bisulphide of carbon (CS;) is a very volatile liriuid, having 

 Ik most dis.igreeab!o odour. It is produced by passing sulphur 

 vapour over red-hot charcoal, and also during the decomposition of 

 various organic bodies. Wo can find no record of this substance 

 being found in coal, or the atmosphere of coal-mines. Carbon 

 bisulphide might bo detected in a sample of air from a coal-mine 

 by the following method : — First free the air, in the usual m.inner, 

 from sulphuretted hydrogen; next pass it in a slow ftream through 

 an ethoriil solution of triethylphosphine, contained in a set of 

 nitrogen bulbs ; a reddening of the liquid indicates the presence of 

 the sulphide of carbon. Experiments of this kind require a know- 

 ledge of analytic methods and manipulation. Jago'a " Inorganic 

 Chemistry, Tlienrotical and Practical," published by Longmans & 

 ('o., at two shillings, will probably suit you. Davis's Geology, in 

 Collin's Elementary Science Series, forms a useful introduction to 

 the study of that science. — G. H. Moktimer.— Laughing gas is made 

 by heating a small quantity of ammoniac nitrate, which splits up 

 into water or steam and nitrous oxide, or laughing giis. The gas 

 may be collected over water. Mix it with air and breathe for a few 



€Jur iftatDrmatical Column. 



EASY LESSONS IN THE DIFFEBEN'HAL CALCULUS. 

 No. IL 



ILLUSTRATING a differential cn-fficienl by the case of a body 

 falling under the action of the constant force of terrestrial 

 gravity, g, wo anppo,«ed the space (») fallen through by a body in a 

 given time (») to be known,— since for the purpose of our illustra- 

 tion it was not necessary to show how s is determined. But it so 

 happens that, in taking this instance to illustrate the intejrnl c.il- 

 cnlus, wo have to consider how » is determined, from what, iu 

 reality, is all that is known in this case. Wo know that the force, 

 i;, Ijoing constant, the velocity gcnerntod in any time U proportional 

 to the time, so that, it v bo this velocity, wo may write v - j(, suit- 

 able units of time and length being taken. (Usually the unit of 

 timo is a second, that of length a foot ; iu which case, p -322 and 



Now, knowing that r = j<, wo may try the same expedient to 

 de ermine the space traversed at the end of any time ( from rest, 

 as wo employed in the inverse problem. Wo may divide up the 

 tim ! into a number of small parts, and Kupposo the velocity uniform 

 dor ng each short interval of timo. Let us see what comes of this 

 experimont. Take t = n r, where t is very small and therefore n 

 very large. Then, at the beginning of the rth interval, the velocity 

 is(r-l)3r, and at tlio end of this interval the velocity is r ^ r. 

 Thus the space traversed in the interval lies between (r- I) g t' and 

 rg t'. Doing this for all the intervals, and adding, we find that tho 

 total space traversed in time n t or < lies between 



r0 + l-r2-h3+ . . . -H(r-1)-H . . . ■^(n-l)]^r• 



and[l<-2-f3+ -t-r + . . . . +n >r' 



or between 



"-^'"-"•^'"' 



or (writing for r its equivalent l+n) between 



lilzJ^jC and ^4rr^''' 



2 2n 2 2» 



The larger n is the smaller is tho second term of each expression. 

 But we may have n as large as we please, and so bring these two 

 expressions as near to each other in value as we please. This 

 means, of course, that the true value of each, when n is infinite, is 

 gj} 



This, then, is the space traversed in time ( by a falling body 

 starting from rest, under tho action of terrestrial gravity. That is, 

 we have established the relation 



using for the purpose what may be regarded as an algebraical 

 artifice — in reality, disguised integration. 



Before showing how this process illustrates integration, let as 

 examine Newton's geometrical way of dealing with a problem such 

 as the above. Note hov: cumbersome h'th procensts are. 



Let the time { bo rcprefentcd by the straight line A B, and the 

 velocity acquired at the end of timo t (from rest), by the straight 

 line B C at right angles to A B. Now suppose A B divided into a 

 number of small equal parts, say into n paHs, each e<iual to M N ; 

 and from all such points as M, N set up straight lines M P, N Q at 

 right angles to A B, and each taken to represent the velocity at the 

 end of the times re|>resentcd by A M, A N, Ac, respectively. Sinee 

 the vclocitv is proportional to tho time, it is obvious that all such 

 points as P and Q will lie on tho straight line A C (for, otherwise, 

 we should not have PM: QN: BC. Ac. :: AM: AN: AB). 



Now. if wo suppose tho falling body to mov,^ during any 

 small portion of timo ropresonted by M N, with tho rolocity 

 at the beginning of that time, represented by P M, the *|iaco tra- 

 vers, d by tho body in that interval would bo rvpn>spnto<l by tho 

 rcetanglo V N ; whereas, if tho falling body moves durinit this 

 interval with tho velocity acquired at the end of it. roprosentod by 

 Q X, tho space traversed will bo represented by the rectanglo « N. 



