608 MR THOMAS STEPHENS DAVIES ON & c . 



NOTE E. 



ON PROPS. XXXV, XXXVIII, p. 590. 



To remove any latent suspicions that may be entertained of the correctness of the deter- 

 mination of the value of p, in consequence of any one of the former equations (29 32) being 

 virtually contained in the other three, it will be desirable to examine the consequences of such 

 an hypothesis in detail. 



On the hypothesis of j = 3, and the fourth equation being contained in the three others, 

 we can find the value q^ + q 2 * + y 3 4 by means of the other three equations ; and if that hypo- 

 thesis be correct, we ought to obtain the same value of this function as is given in (32). 



Put S v S a , S 3 , 4 , for the sums of the first, second, third, and fourth powers of the roots 



of the equation which results from the elimination of y 2 and g 3 from (29, 30, 31). Then it is 

 sufficiently well known that 



6 S t = 3 S l S 3 + 3 S, (S 2 - 2 S*) + Sf 



Moreover, as the origin of co-ordinates is altogether arbitrary in the investigation by 

 which those equations are obtained, we are at liberty to take it so as to fulfil the condition 



This will convert the equation above into 



6S t =3S a 2 , or2S 4 =S 2 2 . 



Substituting in this the values of S 2 and 5 4 from (30, 32), we have 



2Sa m .S(a mPm t }={S(a m p m 2 )}* 



an equation which is manifestly incorrect. 



The same general result might have been obtained without any hypothesis regarding the 

 origin of co-ordinates ; but the expressions would have been more complex, and the examina- 

 tion of the results more troublesome. 



We are hence (even without an actual solution of the several equations) entitled to infer, 

 that the case of three lines only (or p S), as propounded by Dr STEWART, fulfilling the condi- 

 tions of the porism, is inaccurate. Moreover, as a general theorem ought to be true in all its 

 particular cases, it follows that the general statement given amongst the " Theorems " is also 

 inaccurate. 



NOTE F. 



ON PROPS. XL. AND XLII, p. 594. 



The connection between the porismatic and indeterminate proposition is capable of a 

 striking exemplification, by a comparison of these propositions with the equations of the cor- 

 responding porisms. Owing, however, to the already extended space required for printing 

 the present part of the discussion, that exemplification must be deferred, as well as some other 

 necessary remarks on the porismatic proposition. See, however, Note C., which is a case in 

 point. 



