TOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 345 



an odd number, then m . (2m — l) is an odd number, and consequently z and v 2 

 (see prop, 2) have possible values ; therefore the proposed equation has a quadratic 

 factor, x 2 — 2zr + z 2 — v 2 = O, whose co-efficients are possible ; that is, it has 2 

 roots of the specified form; and it may be reduced 2 dimensions lower. 



Case 3. If m be evenly odd, or \m an odd number, then the equation for deter- 

 mining z, has either 2 possible roots, or 2 of the form a ± by/ — 1, (case 2); 

 and v 2 will be of the form c ± dy/ — 1 ; hence one value of the quadratic factor 

 x 2 — 2-ux -f- z 2 — v 2 = O, will be of this form, x 2 — (2a + 2b */ — l) x + ab + 

 CD ^Z— i =0; and another of this form, x 2 — (2a — 2b y/ — l) . x -{- ab — CD 

 y'— l =0; consequently # 4 — 4ar 3 -j- (2ab + 4a 2 + 4Z> 2 ) x 2 — (4oab -j- 4bcj))x 

 -+• a 2 b 2 + c 2 d 2 = O, will be a factor of the proposed equation; and this biqua- 

 dratic may be resolved into 2 quadratics, whose co-efficients are possible, and whose 

 roots are therefore of the form specified in the proposition. 



In the same manner the proposition may be proved, when -fm, \m t -fem, &c. is 

 an odd number; and thus it appears that it is true in all equations. 



Cor. 1 . If v 2 , or y, be positive, the roots of the quadratic fa'ctor x 2 — 2z# + z 2 

 — v 2 = O, and therefore 2 roots of the proposed equation are possible. If y = O, 

 2 roots are equal ; and if y be negative, 2 roots are impossible. 



Cor. 2. If a possible value of z be determined, and substituted in b, c, d, &c. 

 the original equation will have as many pairs of possible roots as there are changes 

 of signs in the equation y m + by 1 "" 1 -j- cy m ~ 2 + &c. ss 0; and as many pairs of im- 

 possible roots as there are continuations of the same sign. 



XVII. General Theorems, chiefly Porisms, in the Higher Geometry. By Hemy 



Brougham, Jun., Esq. p. 378. 



The following are a few propositions that have occurred to me, in the course of 

 a considerable degree of attention which I have happened to bestow on that 

 interesting, though difficult branch of speculative mathematics, the higher geo- 

 metry. They are all in some degree connected ; the greater part refer to the conic 

 hyperbola, as related to a variety of other curves. Almost the whole are of that 

 kind called porisms, whose nature and origin is now well known ; and, if that ma- 

 thematician to whom we owe the first distinct and popular account of this formerly 

 mysterious, but most interesting subject,* should chance to peruse these pages, 

 he will find in them additional proofs of the accuracy which characterizes his in- 

 quiry into the discovery of this singularly beautiful species of proposition. 



Though each of the truths which I have here enunciated is of a very general and 

 extensive nature, yet they are all discovered by the application of certain principles 

 or properties still more general; and are thus only cases of propositions still more 

 extensive. Into a detail of these I cannot at present enter: they compose a system 

 of general methods, by which the discovery of propositions is effected with cer- 



* See Mr. Playfair's paper, in vol. 3 of the Edinburgh Trans. — Orig. 

 VOL. XVIII, Y Y 



