266 Mr Balfour, On certain points [May 19, 



c. If Co meet the focal chord parallel to qq in m, and if CM 

 be the abscissa of 0, 



C8 .CM=Cm.Co = SK .EK, 



which is the same for both ellipses. Hence 8Q + SQ is the same 

 for both ellipses, 



d. To determine pairs of isochronous arcs in the two ellipses, 

 find two eqnal diameters and draw the corresponding diameters of 

 their auxiliary circles, and draw equal chords in the circles parallel 

 to those diameters : then will the projections of. those chords upon 

 the ellipses determine isochronous arcs as required. 



IV. 



The expression given by Lambert (after Euler) for the area of 

 a focal sector SMN of a parabola may be obtained as follows. 



If P be the vertex of the diameter bisecting the chord MN, 

 viz. in r, then SM+ SN =^{8P + PV) and MN=^JSP.PV; 

 and therefore 



^^^i^^M^^=(VSF±VPFr. 



Hence 



{ 8M+8 N+ MN] ^ (SM+ 8N-MN] ^ 



1 2 \-\ — ;^~^1 __ __ 



= {j8P+jpvy-{j8P-Jpvy 



= JPV{68P + 2PV) 



3 

 = -1=^= sector 8 MN, 

 s/A8 



as is otherwise proved by Lambert in 8ectio i. §§ 60 — 63, 



(4) Mr F. M, Balfour, M.A., F.R.S., On certain points in the 

 anatomy of Peripatus Capensis. 



The discovery by Mr Moseley * of a tracheal system in Peripatus 

 must be reckoned as one of the most interesting results obtained 

 by the naturalists of the Challenger. The discovery clearly proves 

 that the genus Peripatus, which is widely distributed over the 

 globe, is the persisting remnant of what was probably a large 

 group of forms, from which the present tracheate Arthropoda 

 are descended. 



The affinities of Peripatus render any further light on its 



* " On the structure caucl development of Peripatus Capensis," Phil. Trans., Vol, 

 CLXiv,, 1874. 1- ' » 



