KDomledge & Selentifie Neuis 



A MONTHLY JOURNAL OF SCIENCE. 



Conducted by MAJOR B. BADEN-POWELL, F.R.A.S., and E. S. GREW, M.A. 



Vol. III. No. 23. 



[new series. 



OCTOBER, 1906. 



SIXPENCE NET. 



An Interesting New 

 Asteroid. 



Approximation of the Mean Dis= 

 tance and Mass of TG. 



By Prof. F. Tarrida del Marmol, C.E. 



TG, the newly-discovered asteroid, is interesting, not 

 only because it widens as far as the limits of the orbit of 

 |upiter the region of the small planets, but, too, as a con- 

 firmation — according to the results of recent observations 

 — of the elegant theorem of Lagrange : " Three celestial 

 bodies, whatever their masses, can move permanently at 

 the angles of an equilateral triangle." [intil now no 

 confirmation was found in Nature of that curious 

 dynamical possibility, the verification of which the great 

 French mathematician left to future generations. Now, 

 the Sun, Jupiter, and TG are precisely three bodies 

 which, according to the observations alluded to above, 

 verify the proposition of Lagrange. 



The articles by Professor Abetti and by Mr. Crommelin 

 which appeared in the August and September numbers 

 of " Knowledge " respectively, strongly tend to conlirm 

 the above, as does the information published in the July 

 issue ; but I wish to remark, with reference to the 

 latter, that there is surely an error in the observations 

 (juoted "of April 22nd combined with those of Feb- 

 ruary 22nd and March 23rd, which show a mean distance 

 a little grcalcr than that of Jupiter." It is easy to prove 

 that this mean distance is not larger, but rather smaller 

 than that of the giant planet, and it is even possible to fix 

 the limits of that distance and, consequently, obtain an 

 approximation of the mass of TG. By the same process, 

 that approximation will be substituted by an exact calcu- 

 lation when, through observation, the exact distance from 

 the Sun of that asteroid will have been established. 



Let (/ and d' be the respective distances of Jupiter and 

 T("i from the Sun, f the time of their revolution, which 

 is identical for both according to observations, M the 

 mass of the Sun, iii that of Jupiter, and ,v that of the 

 asteroid. 



According to the Kcplerian laws, taking the masses 

 into account : — 



f- 



t- 



Or, 



And again. 



d^ 



d' (M + x) = d'^' {M + m) 

 d'-' M + d'^ m - d" M 



.r = |J- {M + m) - M (a) 



<T-.) + "'S;(») 



The formula (b), since x, the mass of TG, is certainly 

 smaller than m, that of Jupiter, shows that the quantity 



in parentheses must be negative, i.c., that — <r, 



or d' ■' < d' 



or d' < d. 



— O.E.D. 



The formula (a) enables us to find limits between 

 which the distance of TG to the Sun is to be fixed. \\'e 

 know that the mass of the asteroid must be greater than 

 nothing and smaller, at least, than the hundredth part of 

 that of Jupiter (or else it would ha\e been seen long ago). 

 We have then the two inequations : 



'LL{M + m) - M> 0(1) 



, {M + m) 



M < ^(2) 

 100 



M, H/,and d being known, the solution of these inequa- 

 tions gives us the following limits (taking d = 483,000,000 

 miles) : 



d' > 482,855,100 



d' < 482,879,250 



which are both very near — but inferior — to the distance 

 from Jupiter to the Sun. 



.\s to the mass of TG, it will be possible to find it 

 with all exactitude with the formula (a) when more pre- 

 cise observations will have established exactly the value 

 of d'. Meanwhile, if we take a value comprised between 

 the limits suggested above, rather nearer to the inferior 



one w-e find for the relation {'-,) the value 0-9990618. 

 The mass, then, of TG would be 



X == (' j (M ; III) - M = 0-9990618 X 330310 

 — 330,000 = 0-103 1 58 

 a result which, in my opinion, shows nothing unlikely, 

 although I am, for obvious reasons, inclined to believe 

 that the calculation, when based on strictly exact </ii/(i, 

 will give a figure still smaller. 



