DR MATTHEW STEWART'S GENERAL THEOREMS. 597 



and each of these several equations will be doubled by the expansion of the 



2v 

 cosines. Whence, for all the lines except the first we have 2 (1 + 2+ ..... v) 



= 2 (v + 1 ) v : or for the whole number of conditional equations 2v(v + l) + v=v(2v + 3). 



We are, therefore, in this case, led to the conclusion, that except v be even, 

 there will be either an indeterminateness or contradiction in the results of the 

 hypotheses of the porism : that is, n if even must be of the form 4 fj. ; and when 

 this is fulfilled, p = /JL (4 p + 3), instead of 4 /JL + 1 as stated in the ' General 

 Theorems.'' 



When fjL =1 or v=2, we have 1 {4 + 3} =7, as before determined in reference 

 to Props. 35-38. 



2. 





The first line will have v + 1 terms 



... second ...... v + 1 ... 



... third ...... v 



... fourth ...... v 



... fifth ...... v-\ ... 



... sixth ...... v 1 ... 



and so on. 



Whence, as there are 2 v + 2 lines, having all the multiple cosines from to 

 2 v + 1, we shall have in all (v + 1) (v + 2) terms. Also, with the exception of the 

 first, they are doubled by the expansions of the cosines, and the entire number of 

 equations of condition will be 



2 (v + 1) (v + 2)-(j> + l) = + l) (2v + 3) 



When v is even, this condition cannot hold in reference to Dr STEWART'S 

 theorem ; for it will give either an indeterminate or an impossible condition, as 

 before. That is, the form =4 /*+ 1 is precluded from the enunciation. 



When v is odd, the condition is capable of fulfilment : that is, when n =4 /j. + 3. 

 In this case, if we denote v by 2 x + 1, we shall have^=(* + 1) (4 * + 5). 



When v =1, the determination agrees with what was found in Props. 24-25 ; 

 for then 1(1 + 1) (2. 1 + 1) =5. 



We have thus obtained the number of equations to be fulfilled for the general 

 forms of n, as well as the general forms of those equations themselves : and have 

 shewn that the number of lines porismatised is erroneously in all the cases, and 

 impossibly in some of them, laid down by Dr STEWART. For the case of n odd, 

 we see that there can be no number of lines porismatised, except = 4/*+3, in 

 Propositions 50-53 : and in 46-49, except n=4 p. 



Particular conditions will lessen this number. In 46-47 this takes place in 

 the two cases in each proposition. 



VOL. XV. PART IV. 7 Y 



