289 



" The theorem which Mr. Graves has announced results in 

 a very simple manner from a formula given for the first time, 

 1 believe, by Mr. Haedekampe, of Hamm, in the twenty- 

 fifth volume of Crelle's Journal, page 180. In a memoir 

 published in the twelfth volume of the Journal de Mathe- 

 matiques, M, Liouville has demonstrated the same formula 

 from geometrical considerations, and has attached a very ele- 

 gant and precise signification to it. As the theorem in ques- 

 tion, regarded under this point of view, is the natural extension 

 to the case of three dimensions of the property (discovered by 

 Mr. Graves), relating to the excess of the sum of the tangents 

 drawn to an ellipse from a point upon a confocal ellipse over 

 the included arc, it will be well to show how this latter pro- 

 position may be established by the method to which I allude. 

 The generalization will be seen to follow without any diffi- 

 culty. 



" Let us adopt the notation of elliptic co-ordinates. The 

 differential of the length of a right line tangent to an ellipse 

 defined by the equation ix = a, will be 



where 2 6 is the constant distance between the foci. Now, if 

 we remember that the second term is precisely the differential 

 of an arc of this ellipse, and if we fix the origin of the v's at a 

 point (P) on the ellipse, it is clear that the above expression 

 will be equally the differential of the mixtilineal line, com- 

 posed of an elliptic arc, one of whose extremities (P) is fixed, 

 and of the right line tangent on the other extremity. Let us 

 now consider a pair of tangents drawn from a point O to an 

 ellipse, and let T, T' be the points of contact, and P, Q two 

 fixed points on the ellipse. Then the differential of the mix- 

 tilineal line O r+ TP will be 



'/(■^)"^V(i^O' 



