38 Analytical Solution. 



pretation has been admissible, the Examiners have naturally- 

 leaned towards that which has been deemed most conducive to 

 public convenience. It is not impossible tbat in so doing they 

 may sometimes have taken a more liberal view, than the rigid 

 letter of the law might be found to authorise, if disputed. Not 

 only the possibility of such a dispute, by appeal to the Supreme 

 Court, but tbat also of diversity of opinion arising between 

 successive Examiners appointed to conduct the Act, and con- 

 sequent alterations of practice at different periods, confusing 

 to the public, equally render it desirable that all such questions 

 should be set at rest. 



"With this 'observation I conclude. To enter more fully into 

 the details of the amendments suggested would be beyond the 

 scope of the present paper, which has already reached a length 

 by no means contemplated in commencing it. My object has 

 been to convey to any intelligent person interested in the 

 subject, whether professional or lay, sufficient information to 

 enable him to understand its general bearing and to appreciate 

 its value, not in the light of a completed measure, but of a 

 system which has so far vindicated by experience its capacity for 

 usefulness, as to require and to deserve all possible exertion and 

 aid towards rendering it more perfect. 



Aet. II. — Analytical Solution to Sir William Hamilton' 's Problem 

 on the Inscription of closed N'qons in any Quadric, by Martin 

 Gardiner, JEsg_., C. J$., Member of the Mathematical Society of 

 London. 



[Bead before the Society, June 2nd, 1869.] 

 To inscribe in any quadric a closed n'gon (or polygon of n sides) 

 whose sides will pass in order through n given points in space. 



Let S = o be the equation of the quadric in reference to four 

 fixed planes ; let k k k k be any n'gon, open or closed, in- 



12 n ntl 



scribed in S whose sides k k , kk, kk pass in order 



12 2 3 n n+1 



through the n given points whose co-ordinates are a /3 y S , 



1111 



a y S, «(3y8. 



2222 nnnn 



Now putting S, S, S for the values of the function S in 



12 n 



which the variable co-ordinates are replaced by those of the given 



