20 Rev. B. Bronwin on the Determination of 



liquid, which was quite clear, gave a much darker tint with 

 the test than any which had previously been obtained in the 

 experiment. It gave a tint about equal to that given with the 

 unfiltered liquid when diluted with its own volume of water; 

 while it (/. e. the last filtered portion) required to be diluted 

 with twice its volume of water to produce the same tint as that 

 given by the reserved filtered portion (a). The liquid now 

 passed through the filter very slowly; it was tested again, 

 when eight more fluid ounces had passed through, with the 

 same result as before, except that the tint was a trifle darker. 

 This experiment sufficiently shows that the effect contem- 

 plated does occur, and that it would be unsafe to trust to the 

 action of a filter to separate oxide of lead from water for an 

 unlimited time. 



VII. Equations for the Determination of the Motion of a 

 Disturbed Planet by means of M. Hansen's Altered Time. 

 By the Rev. Brice Bronwin*. 



T 



HE theory of M. Hansen on Lunar and Planetary Per- 

 turbations, owing to the two times r and t which it con- 

 tains, is attended with many difficulties, and is very perplexing. 

 His results I think are in an advantageous form ; but perhaps 

 they might be obtained more easily by the equations given in 

 this paper, which are referred to the plane of the orbit as if it 

 were a fixed plane, because I have proved that so referred 

 they are true. [See this Magazine for November 1844, and 

 also the Cambridge Mathematical Journal, No. 24.] 

 The equation 



A* , . . 



— = 1 + e cos (y — it) = 1 + e cos it cos v + e sin ir sin v 



is true for the disturbed orbit; h, e, and ir having their known 

 variable values depending on the disturbing force. If /z , e , 

 and 7r be the values of these quantities when the disturbing 

 force is made to vanish, then 



ecosnr=e cos7r -f- / (cosh- de — esin7r dn), 

 e sin it = e sin tt + / (sin n d e + e cos 7r d 7r), 



These values substituted in the above equation give 

 * Communicated by the Author. 



