352 



its concavity, at the point of contact, turned towards the sun. 

 The moon, or any other satellite, may also be regarded as 

 describing, about its primary, an orbit, of which the hodogra- 

 phic representative shall still be a varying circle. 



As formulse which may assist in symbolically tracing out 

 the consequences of this geometrical conception, Sir William 

 Hamilton offers the following transformations of certain gene- 

 ral equations, for the motion of a system of bodies attracting 

 each other according to Newton's law, which he communi- 

 cated to the Royal Irish Academy in July, 1845. (See Pro- 

 ceedings, vol. Ill, part 2, Appendix III. and V.) 



The new forms of the equations are these : 



ni' 



in which p and r are the vectors of position and velocity of the 

 mass m at the time t; p' and r' the two corresponding vectors 

 of another mass m' at the same time ; o- is another vector, per- 

 pendicular to the plane, and equal in length to the radius of 

 the momentary relative hodograph,representing the momentary 

 relative orbit, which the attraction of the mass m' tends to cause 

 the body m to describe ; d, $, S, are marks of differentiation, 

 integration, and summation, and V, U, are the characteristics 

 of the operations of taking respectively the vector and versor 

 of a quaternion. Or, eliminating p and <t, but retaining the 

 hodographic vector t, and using A as the mark of differencing, 

 the conditions of the question may be included in the follow- 

 ing formula, which the author hopes on a future occasion to 

 develope : 



-(m -f Aw)dU(SArdO 



= ^? 



V(AT.SArd?) 



Meanwhile it is conceived that any such attempt as the 

 foregoing, to simplify or even to transform the important and 

 difficult problem of investigating the mathematical conse- 

 quences of the Newtonian law of attraction, is likely to be re- 



