30 REPORT — 1877. 



8. The equations to the tangents at F, D', the extremities of a transversal 

 through D, are 



^ff-h)(f-F)=0, S|(^-A)(/-F)=0. 



These intersect in the point 



q(/-F) _ /3fff-G) _ V (h-H) 



fF - ffGr - hii ' 

 Its locus is the trinodal quartic, 



9. To investigate the three Cayleyan class-cuhics (of § 1). 

 The equation to the chord A'B' is 



ah( ff -G)+ph(f-F)+y(FG-f !/ ) = 0. 



Let t=/+F =y+G=A+H: the tangential coordinates of A'B' are thus 

 related, 



Now, since the equation to the cubic (of § 2) may be written as in § 4, 

 f(r-f) _ ff(r-ff) _ h(r-h) 



y 2 



,2 



Hence the equation to one Cayleyan is 



(p+q+2r) ( } ?x 2 -qY)+ r\p - qy = 0, 



where p, q, r represent ap, bq, cr ; a, b, c, being sides of A B C. 



The chord C 'D' is also a tangent to this Cayleyan. 



Similarly the equations to the other two Cayleyans may be written by symmetry. 



10. Tbe loci of the several points in which concurrent transversals are divided 

 Arithmetically, Geometrically, and Harmonically, form a system of cubics syzyge- 

 tically connected with the primitive ; and all pass through the tetrad of which the 

 point of concurrence is their tangential point. 



If m 3 =0 be the cubic, and w a its pole conic, the equations to the loci are in piano 



2u 3 — ru 2 = ; u 3 — ru 2 = ; u 2 = 0. 



In spherics the equations must be modified : — 



2m 3 (6V) 2 — ra 2 2rt(«a+6/3 cos c+ey cos6) = 0, 

 m 3 (6V) 2 — rw 2 2a(«a+6/3 cos c-\-cy cos b) =0. 

 u 2 =0. 



The factor of w 2 denotes the quadrantal polar of (D) (r, r, r) the centre of the 

 inscribed circle. 



These results might be anticipated by projecting the plane cubic and its associ- 

 ated cubics on the sphere, D being the pole of gnomonic projection. 



[This memoir has been published in extenso with additions, since the meeting, 

 in the ' Quarterly Journal for Mathematics,' vol. xv. pp. 198-223.] 



On a Method of Deducing the Sum of the Reciprocals of the First 2 p n Numbers 

 from the Sum of the Reciprocals of the First n Numbers. By F. G. Landon, 

 M.A. 



