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of the small circle on a sphere, which osculates at a given point 

 T, to a given spherical conic. Let the given cyclic arcs be 

 ACt AC, extending from one of the two points A of their 

 own mutual intersection to the tangent are CTC, which is 

 well known to be bisected at the point of contact T. On the 

 normal arc NTP, drawn through that given point T, let fall 

 a perpendicular arc AN \ draw NC, or NC\ and erect €P or 

 CP, perpendicular thereto, and meeting the normal arc in P: 

 the point P, thus determined, will be the pole, or spherical 

 centre of curvature, which was required. 



Sir William R. Hamilton communicated a notice by Pro- 

 fessor Young, in continuation of a paper by the same author, 

 on the sum of eight squares, read to the Academy on 14th 

 June last. (See Proceedings, Vol. III., p. 526.) 



The principal object of the author is to shew that the for- 

 mula for eight squares, as printed in the part of the Proceed- 

 ings just referred to, does not admit of extension to the case 

 of sixteen squares, or to any of the more advanced forms. The 

 manner in which the proof of this is conducted may be briefly 

 described as follows. As stated in the former abstract, the 

 construction of the eight-square formula was suggested by a 

 certain law of formation observable in that for four squares. 

 It was under the guidance of this law that the component parts 

 of the more advanced form were constructed and connected 

 together ; thus presenting, when completed, the eight rows 

 of binomials which appear in the before-mentioned abstract, 

 and which, from their construction, are necessarily such that 

 if the quantities composing any two binomials in a row are 

 each made zero (which is equivalent to reducing the eight 

 squares to four), the pre-established four-square formula re- 

 sults. 



It is easy to see, if the sixteen-square form existed, that it 

 would necessarily involve the subordinate form for eight, ex- 



c2 



