482 REPORT — 1880. 



through its point of contact, the above locus must pass through the points of 

 contact of the tangents to the conic from the given points. This theorem enahles 

 us to wi'ite down the equation to the known conic passing through two given 

 points, and the points of contact of tangent from those pomts to a given conic. 



To return to the particular case of the du-ector circle. The tangents from 

 the circular points at infinity intersect in the foci : the points of contact must 

 therefore lie on the polars of the foci, i.e. on the directrices. Heuce the director 

 circle of a conic passes through the intersections of the directrices with the conic. 

 In other words, the directnces of a conic are the chords of intersection of the conic 

 loith its director circle. 



Their equation is therefore of the form 



V — fj-u = (G). 



The conditions that this should represent parallel straight lines are foimd, after 

 rejecting a factor 0, F or G, each to reduce to 



jx- — fj. (a + b) + (ah - h") = 



the quadratic (A) obtained for X. 



I have identified (G) with (B), only X and n must be taken as difierent roots of 

 the quadratic A. 



12. Note on the Shew Surface of the Third Order. By Professor 

 H. J. S. Smith, M.A., F.B.S. 



13, On a hind of Periodicity presented by some Elliptic Functions. 

 By Professor H. J. S. Smith, M.A., F.B.S. 



14. On Algebraical Expansions, of which the fractional series for the cotangent 

 and cosecant are the limiting forms. By J. W. L. Glaisher, M.A., 

 F.B.S. 



The expansions in question are : — 



1 1 1 



a.-(P-.i-) (2'^-.r2) . . . (M^-.r^) n\n\ .v 



1 I 1 1_\ 



(m-1)! (m + 1)! \x - \'^ .L-\) 



1 / 1 1 \ 



"^ (m-2)! (m + 2)! \.i-2'*".r +2/ 



r ] / \ \ \ 



(n — r) ! (m + ?•) ! Vi' — r :v + r) 



n I / \ 1 \ , 



(12 - 2V) (3= - T-.v") . . . |(2w - 1)- - 22^2 1 



.r (1^ - X-) {2"~ - x") . . . iji' - x~) 



"""^^- 1 



(2w) ! (2w) ! 



1 I 2 



\ nl n\ y 



