78 Mr. Sylvester's Analytical Development 



Cor. (5.) When <^ or i^^ = 



And .*. ^ assumes the form — -, which indicates that the ex* 



tremities of the four prime radii are singular points. 



In concluding for the present it behoves me to state that one 

 step has been omitted in the foregoing paper, viz. the actual 

 performance of the eliminations which lead to the rectilinear 

 equation to the wave-surface. But Mr. Archibald Smith's 

 elegant and brief Memoir in the Cambridge Philosophical 

 Transactions of last year leaves nothing to be desired further 

 on that head. 



That I have not exhibited it in its proper place (prop. 6.) 

 arises only from my respect to the principle of literary pro- 

 perty. With this important blank supplied the Analytical 

 Theory may be pronounced to be complete. 



For all errors and imperfections in what precedes my ex- 

 cuse must be press of time and a total want of the materials to 

 be derived from consulting works of reference. 



Since writing the above I have had an opportunity of reading the 

 paper of our living Laplace inserted as part of the Third Supplement 

 to his System of Rays in the Transactions of the Royal Irish Aca- 

 demy, in which the principal foregoing results are obtained by aid 

 of a more refined and transcendental analysis. 



The nature of the four singular points is there discussed and the 

 existence of four circles of plane contact demonstrated. 



The former may be very easily shown thus : when i, is very small 

 i^^ = 2 e — i< cos ^ very nearly ;// denoting the inclination of the plane 

 in which e is reckoned to the plane in which l^ is reckoned. 

 Hence 



= |_(«.-,cO-^(^-^)cose 



2 

 1 1 



b'^ b^ac 



-/(a^ _ 62) {b^ - c^) . (cos yp±\)ii 



.... = i.(l + l(cos^±l)(,-|!)^(5-,)*,) 



Take ;// constant and let the abscissae and ordinates be reckoned re- 

 spectively along and perpendicular to the prime ray. 



