10 REPORT— 1870. 



= {2 «(«—!) (rt— 2)+A} B D F, where a, A are the order and class of the curve 

 a = c—e. In the reciprocal case, where the side-curves are one and the same curve, 

 say B=D = F, we have of course a like formula, viz. the number of triangles is 

 here = {2B(B— 1) (B — 2)+J} ace, where B, b are the class and order of the 

 curve B=D=F. The last and most difScult case is when the six curves are all of 

 them one and the same curve, say a=c=e=B=D=Fj the number of triangles 

 is here = one-sixth of 



. . . . + 1), 

 2a'- ISaH 52a- 4G) 

 -18rt^-|-162a2-420rt+221) 

 52a3 -420a2 + 704« + 1 72; 

 lrt'-40«3+221a=4-172a 



A^. . . -91 



+ A ( . . -12«-|-13o , 

 -9«2+13o«— cool 



vrhere « is the order, A the class of the cm-ve ; a is the number, three times the 

 class -j- the number of cusps, or (what is the same thing) thi'ee times the order 

 + the number of inflexions. 



On a Correspondewe of Points and Lines in Sj^xice. 

 Bt/ Professor A. Catlet, LL.D., F.B.S. 



Nine points in a plane may be the intersection of two (and therefore of an infi- 

 nite series ol) cubic curves ; say, that the nine points are an " ennead : " and simi- 

 larh' nine lines through a point may be the intersection of two (and therefore of an 

 infinite series of) cubic cones ; say, the nine lines are jin ennead. Now, imagine 

 (in space) any 8 given points ; taking a variable point P, and joining this with the 8 

 points, we have through P 8 lines, and there is through P a ninth line completing 

 the ennead ; this is said to be the corresponding line of P. We have thus to any 

 point P a single corresponding line through the point P ; this is the correspond- 

 ence referred to in the heading, and which 1 would suggest as an interesting sub- 

 ject of investigation to geometers. Observe, that considering the whole system of 

 points in space, the corresponding lines are a triple system of lines, not the whole 

 system of lines in space. It is thus, not any line whatever, but only a line of the 

 triple system, which has on it a corresponding point. But as to this some explana- 

 tion is necessary; for starting with an arbitrary line, and taking upon it a point 

 P, it would seem that P might be so determined that the given line and the lines 

 from P to the eight points should form an ennead, — that is, that the arbiti-ary line 

 would have upon it a corresponding point or points. 



The question of the foregoing species of correspondence was suggested to me by 

 the consideration of a S3-stem of 10 points, such that joining any one whatever 

 of them with the remaining nine points, the nine lines thus obtained form an 

 ennead ; or, say, that each of the 10 points is the " enneadic centre " of the remain- 

 ing nine. I have been led to such a system of 10 points by my researches upon 

 Quartic surfaces ; but I do not as yet understand the theory. 



The small oscillations of a Particle and of a Rigid Body. By Egbert 

 Stawell Ball, A.M., Professor of Applied Mathematics ami Mechanism, 

 Eoyal CoUerje of Science for Irelaiul. 



I. Introductory. 

 Laplace investigated the small oscillations of a particle on a sphere, Poisson 

 solved a special case of the same problem on the ellipsoid, Lagi'ange discovered 

 the general laws of small oscillations, and his methods have been improved by 

 Messrs. Thomson and Tait ; the results of which the fallowing is an abstract, have 

 been obtained by a imion of the method of Lagrange in its improved form with 



