1879.] proof of LambeH s theorem. 263 



(iii) That the areas of the sectors 8QQ' and Sqci are iu 

 the subduplicate ratio of the latera recta of their ellipses. 



(i) By the property of ordinates and by parallels, 

 qd" : KU = oH.oP: KP'= OP . OP' : CP' 



= Q0': CD'; 

 and since KL is equal to CD, therefore Q and qo are equal, or 



QQ' = qq'. 



(ii) From above 



SK : GN= C8 : CA= C8 : PIC 

 And by parallels, if CM be the abscissa of the middle point 

 of QQ', 



CN : CM=CP : GO = PK : Ko, 



Therefore SK : CM=CS : Ko ; 



or in terms of the eccentricities e and e, 



e.CM=e .Ko. 



Hence 



SQ + 8Q' = AA' + 2e.GM=PH+2e'.Ko 



= 8q + Sq. 

 (iii) The elements of area which the equal ordinates QQ' 

 and qq in any two consecutive positions cut off from their ellipses 

 are as the breadths of those elements. 



Let a chord through 8 meet QQ' and the tangent at P at 

 rio-ht angles in m and n, and let the breadths of the elements be 

 estimated on 8m and So respectively ; thus it appears that the 

 elements are in the constant ratio of Sm to So, and hence that the 

 whole segments QP' Q and qSq are in that ratio. 



But this is also the ratio of the triangles SQ(^ and Sqq on equal 

 bases QQ' and qq. 



Therefore the elliptic sectors SQQ' and Sqq are as Sm to So, 

 or as Sn to SP, or as CB to CD ; that is to say, they are in the 

 ratio of the minor axes, and therefore (Pi7 being equal to AA') in 

 the subduplicate ratio of the latera recta of their ellipses. 



It follows that the arcs QP'Q' and qBq are described in equal 

 times; or in other words, that the time in the arc QP'Q' depends 

 only upon the lengths AA', QQ', and SQ + SQ', as was to be 

 proved *. 



At the outset we supposed one of the two isochronous arcs to 



* It needs some consideration to see that the theorem is now proved ; but by 

 following out the line of argument briefly indicated in the next paragi-aph it will 

 appear that the proof is in reality quite general. 



Vol. III. Pt. yi. 19 



