118 On the Formation of the Tails of Comets. 



for the effect of its change of direction. For greater simplicity, 

 let us confine our attention to the hour and a half immediately 

 preceding the perihelion passage. In this interval three fourths 

 of the angular velocity of rotation which obtains at the perihe- 

 lion is received, and thus three fourths, or nearly so, of the velo- 

 city of rotation at the same point. We will, in the first place, 

 conceive the force to act continually with its average intensity, 

 and, at the same time, the motion of revolution to be uniform at 

 its average rate ; then, the resultant of the increments of velo- 

 city imparted in the different instants of the interval considered, 

 will be to the sum of these same increments as the chord of 90° 

 is to the quadrantal arc, as \/2 to 1.5708, as 0.89 to 1, and it will 

 be inclined 45° to the axis of the parabolic orbit. Next, to solve 

 the actual case, we must seek for the law of variation of the 

 supposed repulsive force of the sun. Let the force in question 



dv 



be denoted by q>, and we have 9=37- Now the motion of ro- 

 tation is constantly the same as that of revolution, and thus the 



1 2dr x 



angular velocity of rotation =v= — . Hence dv— — —^-. We 



— rdr 



also have, from the parabolic orbit, dt——j =-, — , (the time 



v2m(r-U) v 



being reckoned from the aphelion.) Whence 9= — -^- — - 



) r 3 Xrdr 



/ — \/r — D 

 = v Sm We learn from this expression that the force 



<P equals zero at the perihelion, and attains to its maximum value, 

 41£° from the perihelion, where r=|D: also that the variation 

 of its intensity is according to a more rapid law than that of the 

 inverse squares. It will be seen, therefore, that its entire effect 

 during the hour and a half which immediately precedes the in- 

 stant when the comet is at its perihelion, is greater than the ap- 

 proximate value of this, found above, and is in pretty nearly the 

 same direction. We may therefore make use of the approximate 

 value instead of the true, in seeking for the effect in double the 

 interval just mentioned, as the result will be less. Multiplying 

 then the value of v which has been found by 0.89 and by a/2, 

 we obtain the quantity sought ; viz. v = 1214 miles (per second) 

 in the direction of the axis, outwards from the orbit. The sub- 

 sequent action of the force will only slightly diminish this velo- 



