December 8, 1892] 



NA TURE 



'39 



the glowing of the meteor occurs, is not very dense, and that 

 one yet gels the necessary quantity of heat. 



It may be remarked that we can vary all these numbers 

 within very wide limits without fearing any contradiction, so 

 that we may conclude, therefore, that no difficulty in the sug- 

 gested Iiypothesis arises from this point of view. 



1 have now to deduce the formulae I have mentioned above, 

 and it will be seen that these are very interesting. 



If we take m as the sum of two masses revolving round each 

 other in a conic section, V the velocity, and retaining for the rest 

 the customary nomenclature, we have for the parabola 



V^ = >tV . 



cos"^ Vi » ' 



tan i » + Vs lan^ Va 



Whence it follows without further difficulty : 

 VV 



One takes c as the velocity of the earth in its orbit with the 

 radius R and puts the sun's mass and the mass of the earth = I, 

 so that k' = c'i<. If we consider further that the expression 



sin V2 » [ I - 7s -f'*"^ V* ^] 



can attain the maximum value ^ it follows that 



M> ^ -W-^-'- 

 4x^2 \c) K 



or if c be given in solar days 



■0 009 I 23 



<^): 



(4) 



To apply this to the Nova we must remember that ->15 



c 

 because the orbital velocity may be greater than that in the line 

 of sight. Besides, more than two months have passed since the 

 supposed grazing of the bodies look place, which time must 

 coincide closely with that of perihelion,' up to the time that we 

 have still spectrum ob-ervaiions in hand. Thus t is much 

 greater than 60. Formula (4) — 



/*> 14779 ^ sun's mass 

 gives thus a limit which supposes masses far too small. In 

 reality we might parhapj assume double this without challenging 

 c intradiction. 



The consideration of a hyperbolic movement takes a similar 

 though less simple form. 



If V(, represeais the velocity at an infinitely large distance, we 

 have 



V^ - Vo^ = ^^., 

 , r 



and according to the Theoria Motus — • 



r * - cos F 



cos F 



e tan F - log tan (45'' + 1/2 F) = '^^^ 

 from which it is found at once that — 





W{. 



V^v3£A, 



si 2) \C 



(5) 



cos 

 jsF 



/ e tan F - lo^ tan (45"' + 1/2 F) ' 



The expression for X, if one allows F to vary from 0^-90°, first 

 'k-ctcases, reaches a minimum, and then increases to infinity. 

 I he minimum value can easily bi determined for then 



' , ' ^'" . uxi. [' '^" ^ ~ '°e tan (45° + 1/2 I*")] must be = i. 



^ \€ -~ Cos V ) 



This equation can be easily solved [or special values of f. 

 For the theoretical calculation which is requisite, I have em- 

 ployed another proceeding, as I have already computed the 

 serial values of X for a special value of e, as the following table 



shows : — 



F = 4 

 8 

 12 

 16 

 20 

 24 

 28 



32 

 36 

 40 

 44 

 48 

 52 

 56 

 60 

 64 

 68 

 72 

 76 

 80 

 84 

 88 



10-207 

 5-224 

 3614 

 2-852 

 2-429 

 2178 

 2-027 

 I-94I 



1 900 

 1-892 

 1-911 

 »953 

 2-017 



2-I0I 



2 208 



2 •34* 

 2-510 



2 72S 

 3026 

 3477 

 4-308 

 6-991 



14393 

 7-302 

 4-987 

 3-866 

 3-226 

 2-827 

 2-569 

 2 400 



2-293 



2-234 



2-211 

 2-220 

 2-257 

 2*323 

 2-420 

 2-552 

 2-729 

 2968 



3'307 

 3830 

 4-802 

 7938 



43071 

 21-689 

 14 630 

 11 156 

 9-118 

 7802 



6 902 

 6-265 

 5 810 

 5486 

 5 266 

 5-'3i 

 5-072 

 5086 

 5>75 

 5-351 

 5-635 

 6070 



6739 

 7843 

 9-991 

 17091 



For very large values of e the minimum of X occurs if 

 sin F = \/2/3, 

 and the minimum value of X becomes 



Min X :r ^ /3i'i:f 



V 2 



1612^/^ 



(6) 



But one practically commits no error if one employs (6) also for 

 the values of e nearly equal to i, as is evident from the following 

 computation of the minima values taken from the above table, 

 and calculated according to foi nulla (6). 



v,;-i 



Jt:-)" 



(7) 



For the above assumptions — 



'=^o,(l)=30, 



we find 



which formula holds good for values of e, which do not quite 

 equal I. In order to include also the parabola we suppose 



fl> 1 5000 ^,/^ 



r 



V- 



(7«) 



NO. 1206, VOL, 47] 



Thus in this case we result in extremely large masses, which are 

 not very probable, or we must assume that " = very nearly i. 



Even for _•* — 0-9, according lo the above formula, /u > 1200 i^c, 



and we may consider the above-given assertion as justified. It 

 has already been remarked that ihis suggested inequality proves 

 only that )U is very much greater than the right side (of the 

 equation). 



It is easy to find a higher limit for;u if ~" does not differ much 



from unity. 



■y* - V "^ 



If we put -— _° - = v, we obtain cos F — ve^ and according 



V + Vq 

 to formula (5) : 



\l -f !»/ \c I K " o* tan F - »'loj^un(45"-t- 1/21-/ 



