576 MR THOMAS STEPHENS DAVIES ON 



matic points, porismatic lines, etc. ; and that these points, lines, etc., are said to be 

 porismatised, instead of given, as usually expressed. 



The arbitrary point is invariably denoted by the polar co-ordinates r 6 ; the 

 porismatic are denoted by the unknown co-ordinates of the point, if a point be 

 porismatised ; and by the equation of a locus with unknown co-efficients, if a line or 

 any other locus. The equation of the porism is then formed by means of these 

 co-ordinates of points, or equations of lines, the several data of the proposition, 

 and the arbitrary point r Q* Then, since r is perfectly arbitrary, the general 

 equation of the porism can only be fulfilled by the co-efficients of the several 

 combinations of r and which appear in it, being separately and simultaneously 

 equal to zero. This equating to zero of those several co-efficients, gives a num- 

 ber of conditional equations, involving the several porismatic unknowns ; and we 

 must have as many equations, independent of each other, as there are unknowns 

 porismatised in the statement of the proposition. Should the number of these 

 conditional equations be in excess or defect of the number of porismatised 

 unknowns, the porism is incorrectly stated. However, it is always easy to cor- 

 rect the enunciated porism so as to fulfil these conditions, either by abstrac- 

 tion from the number of porismatised entities, or by addition to them, as the case 

 may require. 



The number of conditional equations may, however, be correct, and yet the 

 porism not true : for if there be not corresponding real values for each of the 

 unknowns deducible from these equations, it will follow that the conditions 

 of the porism are inconsistent with each other. The complete algebraical solu- 

 tion of a porism requires, therefore, that the conditional equations shall be either 

 actually resolved, or at least that it shall be shewn that the roots of the final 

 equations in each of them, from which all the others have been eliminated, are real. 



In the first part of this discussion, I have, in the main, attended to the for- 

 mation of the conditional equations of the porisms, and the correct determina- 

 tion of the number of porismatic points and lines : but still I have resolved the 

 equations themselves in a great number of cases, including those belonging to 

 several porisms that have not been before established by any method. The 

 equations, however, that result are of a peculiar class and admit of easy dis- 

 cussion by one general method. The preliminary discussions which force them- 

 selves upon us in the solution of these, would occupy so much space, that I 

 have thought it better to defer them to the second, or concluding part of the 

 paper. I have, however, judged it proper to give, in one case, a separate 

 proof of the erroneous number of porismatic entities, enunciated by Dr STEWART, 

 in order to remove any latent suspicion that certain of the equations were vir- 



* When the point is not entirely arbitrary (as in most porisms is the case), r and 6 will be con- 

 nected by an equation which defines the locus of the partially arbitrary point. Any detail upon this 

 head would, however, be altogether irrelevant in this place. See " The Mathematician," as above. 



