DR MATTHEW STEWART'S GENERAL THEOREMS. 005 



I form of the abilities of the author, as evinced by his varied correspondence with the most 

 valuable English periodicals devoted to mathematical researches. 



My old friend and former pupil, Mr LESLIE ELLIS (in the Cambridge Mathematical Jour- 

 nal, May 1841), has proved a considerable number of the indeterminate theorems which 

 relate to inscribed and circumscribed polygons, by means of a very remarkable theorem 

 (which he has there investigated), which reduces the sums of the specified powers of the lines, 

 or perpendiculars, to a definite integral. The following is his theorem ; but, for the manner 

 of employing it, reference must be made to the paper itself. 



In a letter to me, Mr ELLIS also suggests the application of the same process to sphe- 

 rical polygons. It may also, evidently, with slight modifications, be applied to regular poly- 

 hedrons, inscribed and circumscribed to a sphere. 



Should any other English authors have discussed these theorems, their works are unknown 

 to me, and that, after taking much trouble to discover all that had been attempted relative to 

 them. I am not aware of any writer on the continent who has distinctly dwelt upon them, 

 except M. CHASLES, in his Aperfu Historique des Methodes en Geometric, though LHUILLIEE, 

 CARNOT, and many other distinguished continental geometers have made occasional reference 

 to SIMSON'S Porisms and STEWART'S Theorems. Both works are, however, by them con- 

 sidered merely as relating to indeterminate theorems. CHASLES forms the highest estimate of 

 STEWART'S researches ; but as his view of the ancient porism is so opposed to Dr SIMSON'S, 

 he was not likely to be led to any method of investigation adapted to the discussion of these 

 propositions : in fact, he does not offer any. He, however, enunciates the " extension " of 

 two of the general theorems (pages 353-54), namely, the 44th and 53d. These extensions 

 will be true only when the original theorems respecting the numbers of porismatic points 

 and lines, as given by Dr STEWART, are corrected. In this case, all the equations of condi- 

 tion for different values of 8 are contained amongst those for 8 ; and hence the porismatic 

 lines, which fulfil the conditions for the value = 0, fulfil those, also, for all higher values 

 within the prescribed limits ; although the conditions of the porisms might be fulfilled with 

 a smaller number of porismatic points and lines for those cases. 



CHASLES'S extensions are, in our notation, 



the former being that of Prop. 44, and the latter that of Props. 49 and 53, as n is odd or even ; 

 and d being taken from 0, 1, 2, .... jj in the former case, and from 0, 1, 2, ^ in 



the latter. 



Finally, it may be remarked that nearly all the general theorems have analogous ones in 

 respect to points, lines, or planes situated arbitrarily in space, the numbers of porismatic ones 

 being properly chosen. Several of them, too, have corresponding properties (with a different 

 porismatic number, of course), with respect to the hyperbolo'id of one sheet ; but this is not 

 the proper time for details on any collateral subject. 



VOL. XV. PAKT. IV. 8 A 



