OP ARTS AND SCIENCES : NOVEMBER 8, 1865. 43 



If now Y be the spherical circle at infinity, and W two separate 

 or coincident points, we see that whe^i U has double contact with V, 

 or envelopes it, so does every homqfocal to U some homofocal to V; 

 and all the planes of contact intersect in one line or point. 



C/'may have 2, 3, or 4 double contacts with the surfaces V-\- y^Y, 

 since the condition 



I' U-\- m ' V-\- m iJL'Y = product 



is equivalent to three quadric conditions among (I, m, m ^i), which are 

 satisfied in from to 4 ways, just as three conies have from to 4 

 common intersections. 1 of these double contacts may be replaced by 

 1 envelopment ; or all 4 by 2 envelopments if (U, V) be cones. If 

 U have p double contacts and q envelope-contacts with surfaces V -j- /n Y, 

 so has V with surfaces U -|- X Y. 



Of the fact that homofocals envelop homofocals, a familiar case is 

 that each focal line of a cone U touches either co-planar focal of any 

 enveloped quadric V ; whence the circularity of that enveloping cone 

 whose vertex is on a focal ; and the consequent linear relation among 

 the four distances of two points on one focal from two on the other, &c. 

 Other known cases are, that Z7's focal lines, when not thus co-planar, 

 are generators of some homofocal to V ; and that when Z7is a quadric 

 of revolution, its foci lie upon F's focals. For in neither of these 

 three cases could the focal curve of U otherwise envelop any homo- 

 focal of V. 



P. S. Feb. 10, 1866. — The above was ia the printer's hands, before I was 

 aware how much of it had been given by Salmon ; but I retain it with some 

 changes, as certain points in it may still be new. — J. E. O. 



The following gentlemen were elected members of the 

 Academy, viz. : — 



J. Victor Poncelet, of Paris, was elected a Foreign Honor- 

 ary Member, in Class I., Section 4, in place of the late M. 

 Struve. 



Mr. Lewis M. Kutlierfurd, of New York, was elected Asso- 

 ciate Fellow, in Class I., Section 3. 



Mr. Samuel Eliot, of Boston, was elected a Resident Fellow, 

 in Class III., Section 3, and Mr. G. W. Hill, of Cambridge, a 

 Resident Fellow, in Class I., Section 2. 



