OF ARTS AND SCIENCES : NOVEMBER 8, 1865. 39 



of about three hundred and twenty stars is now in process of reduction, 

 which depends on nearly ten thousand observations made since 1862. 

 The instrument has shown a stability of mounting sufficient for an inde- 

 pendent and accurate determination of thirty-five polar stars, while it 

 has also proved itself capable of being used for exact right-ascensions 

 of time-stars by the excellence of its pivots, and its general stiffness. 



The catalogue will be, if I am not mistaken, the first American 

 Fundamental Catalogue of Right- Ascensions ; I can venture to call it 

 so, although the still unfinished normal clock, and the lack of a good 

 circle for solar observations, compel some reliance upon the general 

 accuracy of other determinations. But with the requisite apparatus, 

 about one thousand more observations would free us from even this 

 necessity. 



Mr. Oliver presented the following paper : — 



On some Focal Properties of Quadrics. By J. E. Oliver. 



I. Any two quadrics have one, and usually but one, common auto- 

 polar tetrahedron, T. Referred to this their equations become 



U= aw 2 + k 2 -fcy 2 -(- dz 2 = 0, 

 Y = a W 2 -f /3z 2 + yf -f- 8 z 2 = 0, 



whether in tangential or in point-coordinates. Using tangential coordi- 

 nates, all the quadrics 



U+\Y= (a + Xa) w 2 + = (1) 



have a common enveloping developable ; and the entire system is 

 determined by any two of its quadrics [or by any eight of its 

 developable's planes which are not specially related]. 

 The four quadrics that correspond to 



X- - a A - - ^ A - C 1A- - ^ 



a /3 y 8 



are plane conies in the respective planes of reference. 



Deforming T and (1) together till T has one plane at infinity and 

 the other three mutually orthogonal, and then lengthening in suitable 

 proportions the three sets of principal axes, the rectangular point- 

 equation of the system becomes 



* 2 . + £_ + — - — 1, (2) 



S D2 I 7„2 ' /T2 I 1.2 V / 



A 2 -\-k- ' £ 2 + F ' <7 2 -f- k 



