594 MR THOMAS STEPHENS DA VIES ON 



Now, to consider these in relation to the porism, we must find the number 

 of equations which are produced in this series. The forms of them are given 

 above ; but it will be more convenient to recur to the original development in 

 Lemma i., in order to discover this object of our inquiry. 



1. From the terms clear of the 6, we have the alternate terms of a binomial 

 (disregarding the special co-efficients in all cases, which do not affect the actual 

 number of terms) of the 2nih degree : that is, exclusive of the term r 2 " which can- 

 cels in the porism, we have n terms. 



2. In that which involves cos 6, we have the terms which were omitted in 

 the first line, viz. n terms : and a similar number in that which involves sin 6. 



3. In each of those involving cos 2 Q and sin 2 6 we have all those of the 

 first line except the first and last ; or n 1 in each case. 



4. In each of those involving cos 3 Q and sin 3 & we have all those of the 

 second line except the first and last; or n -2 in each case. 



Proceeding thus, we find at each successive step a diminution of one in the 

 number of terms belonging respectively to the cosines and the sines of the 

 multiple arcs : till in the last line we get one term involving cos n 6 and another 

 involving sin n 6. 



Again, the number of lines which involve cosines and sines is the same as 

 the number of multiples of which are involved : that is, there are n lines which 

 give the double of the number of powers of r in the co-efficients. Whence, in- 

 cluding the first line, there will be 2(1+2 + 3+ . .. +ri)+n=n(n + l) + n=n(n + 2) 

 equations of condition involved in the statement of the porism. 



Now, the determination of a point involves two conditions : viz. such as 

 will enable us to find u p and w p . Whence these n (n + 2) equations will require 

 that \ n (n + 2) points should be porismatised, instead of n + 1 as stated by Dr 

 STEWAHT. 



Again, except n be even, the number of conditional equations will be odd; and 

 hence there will not be the requisite conditions for the determination of any 

 number of points either giving one condition too few, which would render the 

 first point indeterminate, or one too many, which may be (and, generally, would 

 be) contradictory amongst themselves. 



We thus see that except n be even, the porism cannot be true, which is a 

 limitation not laid down in the ' General Theorems' : and we see that when n 

 is even, the number of lines to be porismatised is ^ n (n + l)=p, and not n+ 1 as 

 there stated. The relation to be observed between m and n, will be discussed 

 hereafter, when we come to consider the structure and solution of the equations 

 of conditions themselves. It will hence be unnecessary to discuss this question 

 further in this place ; though we may remark, that this result, when applied to 

 a former case (Props. 34, 35), is in keeping with the conclusions there obtained. 

 For, in that case n+1 instead of \n (n + 2), n being equal to 2, or 3 instead of 4 

 points, are stated to be determinable. 



