DR MATTHEW STEWART'S GENERAL THEOREMS. 577 



tually contained in the others. (Note E.) A formal proof of the correctness 

 of my own determination will be contained in the actual solution of the general 

 conditional equations. It will there appear, not only that this determination is 

 correct, but also that, under these modified conditions, the values of all the 

 porismatic entities are essentially real. 



The indeterminate theorems except those established by Dr STEWART, are also 

 proved in the present part of the discussion. As these have been established 

 already in different ways, it was not necessary to dwell upon them at any con- 

 siderable length, the mere indication of the application of our lemmas to this 

 purpose, being deemed sufficient for the object in view. 



Of the few propositions regarding loci, it is sufficient to remark, that this 

 branch of the subject is too well understood at the present day, to need discus- 

 sion here ; and, at the same time, that some of them are true, only under 

 special relations amongst the data, instead of universally true, as Dr STEWART 

 has enunciated them. 



The five properties of the circle which form the concluding ones of the 

 General Theorems, are deferred to the next part of this paper, not from any 

 peculiarity in the manner of treating them, but to equalise, as much as possible, 

 the extent of the two parts into which the discussion is divided. 



In the enunciations, the forms are altered to suit the view under which the 

 method of solution here employed would give them in the most convenient 

 shape for use. An abbreviated notation, too, is employed in the expression of 

 the theorems : but the principle of that notation is so simple, and, indeed, the 

 notation itself is so commonly in use for analogous purposes, as to render it 

 almost needless to distinctly specify it. It is 



! + a + + a m = Sa m , 



a i r l n + a 2 r^ n + a m r m " =S(a m r m n ] ; etc. 



The classification, also, of the General Theorems is here added. 



1. Indeterminate Theorems. 1, 2, 8, 4, 5, 6, 7, 8, 22, 23, 26, 27, 28, 29, 34, 

 39, 40, 41, 42, 45. 



2. Porismatic, relating to points and lines. 9, 10, 11, 12, 13, 14, 15, 16, 17, 

 18, 19, 20, 21, 24, 25, 30, 31, 32, 33, 35, 36, 37, 38, 43, 44, 46, 47, 48, 49, 

 50, 51, 52, 53. 



3. Loci. 54, 55, 56, 57, 58, 59. 



4. Porismatic, relating to the circle. 60, 61, 62, 63, 64. 



A few notes are added on some articles in the paper, as it appeared more 

 convenient to place these remarks in such a form than in scholia or foot-notes. 



ROYAL MILITARY ACADEMY, WOOLWICH, 

 March 14. 1844. 



VOL. XV. PART IV. 7 R 



