Oct. 20, 1882.] 



• KNOWLEDGE 



341 



England associated with the movements of so distant a 

 volcano, but we shall see as we proceed that there is good 

 reason for believing that an association of this sort actually 

 exists. Very few geologists doubt, indeed, that Vesuvius 

 is connected with the region of disturbance we have already 

 referred to as the probable centre whence English earth- 

 wravea have been propagated. 



THE COMET. 

 Br THE Editor. 



(Continued from page 328.) 



IF we are right in concluding that the comet which 

 passe<l its perihelion on Sept IV is the same body 

 ■which was seen in 166S, 1S43, and 1.^80, the inference 

 unquestionably is that the comet will before very long lie 

 destroyed by being absorbed into the sun. Let us inquire 

 how this process of absorption, or such a process, would 

 take place. I find that very erroneous ideas are 

 commonly entertained on this suViject It is supposed liy 

 many that each successive return must of necessity pro- 

 duce' greater and greater effect : and again, that the 

 acti6n by which the orbit is changed from a long oval to 

 a circle is that which must in most marked degree affect 

 the comet's physical condition. But the reverse is, in both 

 instances, the case. Theoretically the effect at each return 

 to perihelion ought to be less and less in the comet itself, 

 and diminish its period and the length of the greater axis 

 of its orbit in less and less degree : and again, the effects 

 produced on the comet's physical condition by solar action 

 will theoretically be enormously greater after the orbit has 

 been changed to the circular form than they are now while 

 the orbit is elliptical 



Let us, in the first place, consider the varying velocities 

 of the comet at its point of nearest approach to the sun, 

 this point — the perihelion — remaining unchanged in 

 position, but the greater axis and period diminishing. 



The formula for the velocity at any point in an elliptic 

 orbit is as follows : — 



v==o(£->) 



where C represents the accelerating force of the sun at a 

 unit of distance, a the mean distance or half major a.vis of 

 the orbit, and /■ the distance of the body at the moment. 



Suppose now that the distance of our comet from the 

 sun's centre, in perihelion, is half-a-million of miles, or 

 roughly, one 18-"ith part of the earth's mean distance, and 

 note that a bod_v mo\-ing in an orbit having an infinite 

 major axis and perihelion 430,000 miles from sun's centre 

 would travel at a rate of 380 miles per second. Thus we 



which our comet, coming originally from a practically infinite 

 distance, reached its perihelion on its first visit to our solar 

 system, we have 



o 

 V^ = C 



wherefore 



\- 



500,000 

 380 ' *^ 



yg = 380^86=355. 



Let us find now a general formula connecting the velocity 

 »' at distance r in a parabolic orbit with v the velocity 

 at distance r in an elliptic orbit, having mean distance 

 a. We have 



- . = c(?-^) 



\r a' 



I- r V /• a/ ia-r 



and 



and so long as c is very small compared with a we have 



11 -kj _ ?( — » r _ r 



f ia — r u 4« 4f7 



This is a very convenient formula in the case of our 

 comet, for we have in this case *' = 352 (miles per second). 

 Thus, suppose the mean distance « reduced from infinity to 

 3,000,000,000 ; then velocity in perihelion (at 500,000 

 miles from sun's centre) is less than this by 

 5 352 



120000 (352) or by o^qoo "'^^^ P®"" second, 

 i.e., by less than ^\th of a mile, or 26 yards per second. 

 This is all the reduction of velocity in or near perihelion 

 required to change the orbit from the parabolic form to an 

 elliptic orbit, having a mean distance rather greater than 

 Neptune's, and a period not very ditl'erent from the time 

 occupied by the comet in returning after its ^-isit in 1668. 

 For, putting 1843 as the time of its next return, and, 

 therefore, 17.t years as its time of circuit, we find the mean 

 distance corresponding to this period to be % '(17.'>)-. Here 

 is a case for the use of logarithms : (for who wants to square 

 175, and then work out the cube root of the result ? Not 

 I, certainly : but) — 



losr. 175 = 2-24304 



3) 4-48608 

 1-49536 = log. 31-287 

 showing a distance 31^ times the earth's, or about 2,910 

 mOLions of miles. 



Next, let us see what reduction would be necessary to 

 reduce the mean distance to 1,000 millions of miles. The 

 same process shows that the loss of velocity is 



— '—- (352) or -^-^ miles, 

 40000 ^ ' 8000 



i.e., about three times as much as in the former case, 

 say 78 yards per second, which means a further loss of 

 velocity at this next perihelion passage of al>out 52 yards 

 per second. This corresponds nearly enough with the 

 period of 37 years, in which the comet seems to have per- 

 formed its next circuit, — between 1843 and 1880. For 

 we get in that case for the mean distance i/'(37)- : whence 

 the sum 



log. 37 = 1-56820 



add log. 03,000,000: 



1 04547 

 ; 7 -96848 



9-01395=log. 1,032,000,000. 

 So that if the comet came out from a practically infinite 

 distance, and were retarded during perihelion passage V»y 

 only 26 yards, it would circuit next in a period of 175 

 years : and if at its next return it lost only 5:2 yards per 

 second mon; it would complete its next circuit in 37 years, 

 or therealwuts. 



But now, what would be its velocity in perihelion, if it 

 travelled once round the sun in 2| years, its perihelion 

 distance of 500,000 miles being unaltered 1 The mean 

 distance corresponding to this period will be^'^V' times the 



earth's. =yrill 



Now log. 7 111 =085193 



Jrd of this =0 28398 



L"g. eartli's distance =7 06848 



6-2.5246 =log. 178,840,000. 



