Motion and Spherical Trigonometry. 

 radius /O will pass through a and d, since 



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Ea.Ed=EL 2 = EO\ 



(1) 

 and 



(2) aOL = EOL-EOtf = ELO-E<ZO = tfOL, 



so that OL bisects the angle aOd, and passes through g on 

 the circle round Oad. 



Then gha, gaO are similar triangles, and gL .gQ=ga 2 , so 

 that L and are inverse points with respect to the sphere, 

 centre g and radius ga ; thence LT : TO is a constant ratio, 

 equal to La : aO, and similarly LQ : QO is the constant ratio 

 LA : AO round the circle AQD, and OQ = OA . A(/>. 



The inverse of the circle aTd with respect to is another 

 circle a'T'd' parallel to A'Q'D ; for LT : OL'=LT : OT, a 

 constant ratio, so that L'T' is constant, L' being the point 

 inverse to L. 



Conversely the inverse of a system of parallels of latitude 

 on a sphere with respect to a point on the sphere is a 

 system of dipolar circles in a plane, as the circles of latitude 

 on the stereographic representation of a hemisphere. 



6. The line / T / from 0', the centre of the sphere on the 

 diameter OD, makes a constant angle, c 1 , with O'L', and the 

 angle DO'T' is double the angle DOT' ; so that if the arc 

 DT in fig. 5 in the representation on a sphere, centre 0, is 

 produced to double length to V, OV will make a constant 

 angle c' with OA, which is parallel to O'L', and the arc 

 AV = c'. 



Fi<r. 5. 



Then in fig. 5, by Spherical Trigonometry, 

 (1) cos AT.cosDT = J(cos AD + cos AV)=-^(cosc+ cose'), 

 a constant ; so that T describes another sphero-conic, interior 



