120 On the Formation of the Tails of Comets. 



J p • xdx -Jj 2 ■ xdx=g\o%. x -^ - -% + C. = - g log. D + 



g D 2 x g x 2 g D 2 



j7 • g- ; whence GY=g-log. g-^ ' "2 +^i * "jf- Integrating 



between the limits a;=D, and x = d, we have, GY = ,g , log. 

 d g d 2 g D 2 d g g D 3 g* d 



P--^ , T+^" 2-=5- l0 §-D-2 + 2'^ andY= G ,l0 S-D- 



' l+i'l-^ (6). NowtofindG; ^ =f^ dx -f^ dx =, 



G^ 2 G 



a; 



^ ft ft 



— -t; #-f C ; and C=y^+-^-D. Substituting and integrating 



g g g 

 between the limits x=D, and x=d, we obtain G=j=v — l^"!^ 



d D 2 d D 2 



-^,(7). Hence Y=____ — =: ^ t p l f 



V~d+d 2l) ~d 2d D d + d 2 rf 

 (8). Making the calculation for the perihelion, (taking the peri- 

 helion distance = 1,) we get Y s= 4.19183 = 2,095,915 miles. 

 Equation (7) gives G=0.989^; g-=weight of unit of length of 

 tail at the unit of distance. Now we find that the sun's attrac- 

 tion for the head exceeds that for the tail in the ratio of 102,042 

 to 1 : thus, dividing Y - 500,000 miles in this ratio, we have for 

 the distance of the point of application of sun's force from the 

 centre of the nucleus, 15.6 miles. The distance of this point 

 from the centre of gravity will then be 24,975-15=24,960 

 miles. Assuming the deviation of the tail from the line of the 

 radius-vector to be as great as 45°, the arm of lever of the force 

 will then be 17,652 miles. Substituting this for p in equation 

 (5), we get v = 13,713 miles. Three fourths of this, or 10,284 

 miles, will then be the velocity of translation due to a force act- 

 ing with the above mentioned arm of lever, continually in the 

 same direction, and of sufficient intensity to impart the additional 

 velocity of rotation received during the hour and a half which 

 immediately precedes the perihelion passage. Now, to obtain 

 the velocity due to such a force acting after the same manner as 

 the sun ; I find, by calculation, that the portion of the latter 

 force which acts upon the tail is about one half less 90° from the 

 perihelion. Since, however, this is only from the T j]jj to the 

 T __i___ part of the force acting upon the head in the different 



