DR MATTHEW STEWART'S GENERAL THEOREMS. 575 



which it can be applied are strictly, in form and essence, porisms. At the same 

 time, I saw that, in most geometrical porisms, the co-ordinate method would 

 give considerable facility in conducting the actual solution ; and having applied 

 this method to a considerable number of porisms which had been treated geo- 

 metrically, its application to Dr STEWART'S general propositions became natural. 

 In this way, by the use of rectangular co-ordinates, nearly the whole of the pro- 

 positions Avhich had been discussed by Dr SMALL, and Messrs LOWRY and SWALE, 

 were readily established, together with the last five porisms of Dr STEWART 

 respecting the circle. A few of these were sent to a periodical work ; but some 

 circumstances connected with that paper, induced me to lay aside the subject 

 altogether, till a recent period. The views to which I was at that time led, have 

 been since explained in the " MATHEMATICIAN " (Nos. 1 and 2), to which I must 

 refer for details Avhich would be unsuitable to the present paper. I had inten- 

 tionally omitted from this latter paper all reference to Dr STEWART'S theorems, for 

 two reasons -.first, That I had found the insufficiency of the rectangular co-ordi- 

 nate system to meet the object of the more general propositions, from its always 

 giving a redundancy of conditional equations, arising out of a peculiarity in the 

 expressions ; and, secondly, that I had found the method of polar co-ordinates free 

 from this embarrassing objection in all the cases I had tried, and hoped to find it 

 so in all cases whatever. Having now found that such is the case, and having 

 likewise discovered a method of solving the equations to which Dr STEWART'S 

 porisms give rise, I am desirous of laying the results before the Royal Society 

 of Edinburgh. 



Many reasons induce this wish. Dr STEWART'S position in the University 

 of Edinburgh, and his being one of the most distinguished of the original Fellows 

 of the Royal Society, are reasons, hoAvever, paramount to all others ; and I am 

 led to believe that an interest \vill be felt (even in a subject purely relative to spe- 

 culative mathematics) by that Society, to which I ought to pay respect. Another 

 is, that the polar equation of the straight line, of Avhich so much use is made 

 in this discussion, was first given, incidentally, in the Edinburgh Transactions 

 (vol. xii.) ; and the present is the first application made of that system of equa- 

 tions, except to comparatively elementary inquiries. The subject of these equa- 

 tions has, however, been more amply developed in my recent edition of Dr 

 HUTTON'S Mathematics (12th edit.), to which reference may be made in any case 

 Avhere the first sketch, already referred to, may be considered incomplete. 



Adopting, as I do, without modification, Dr SIMSON'S definition of the poris- 

 matic proposition, and taking into account that the point from which lines are 

 draAvn (either to points or perpendicular to lines), is arbitrary, the following 

 statement of the process which I employ will appear both simple and obvious. 

 It will, however, be necessary to remark, that the points, lines, or other entities, 

 which the proposition affirms to be determinable, are called, for precision, ports- 



