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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 formule 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. 11], part 2, Appendix III. and V.) 
The new forms of the equations are these : 
a (8 mi’ 4 
V-(p'—p)(r’—7)’ 
in which p and r are the vectors of position and velocity of the 
mass m at the time ¢; p’ and 7’ 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 
piss hedis se = 7 = SfodU(p’—p); 
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, §, =, 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 o, but retaining the 
hodographic vector 7, 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: 
s¢(m +:Am)dU GArde) - 
V (Ar. Arde) 
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- 
