DR MATTHEW STEWART'S GENERAL THEOREMS. 535 



Insert these in the equation of the porism (putting the expansion in multiple 

 cosines), cancel, and equate to zero the co-efficients of cos 2 and sin 2 0. Then 

 there results, 



2 S (a., cos 2 0_) 

 cos 2 w, + cos 2w 2 = - , 



s a m 



2 S (a m sin 2 m ) 

 sin 2 co, + sin 2 Wo = - . 



tia m 



The solution of these equations gives 



Cos2w = ( m cos20 m )R . S(a m sin28 m ) 



S (a, cos 2 m ) =F R . S (g m sin 2 m ) 



COS .J Wo = F; ; 



A a m 



g _ (S ,)* - {-S 1 (, cos 2 g m )} 2 - {g ( M sin 2 m )}' 

 {S (a m cos 2 e m )} a + {S (a, sin 2 ro )} 3 



Since the angles are symmetrically involved in the general expression, we 

 see that the double sign does not imply different possible solutions, but merely 

 that the sign of cos 2 w 3 depends upon that which we select as belonging to 

 cos 2 eo r 



That it is always real, is at once obvious : for the denominator of the ra- 

 dical is essentially positive ; and the numerator is convertible into 



2fl 1 a 2 {l-cos2(<9 1 - 



3 {l-cos2(0 2 - 3 )}+2a 2 a 4 {l-cos2(0 2 -0 4 )}+ ____ 



+ 2a m _ 1 . a m {l-cos2(0 m _ 1 -0 m )} 



which is, also, obviously positive. 



Again, the expressions for the single angles w t , w 2 are found from the pre- 

 ceding (3, 4), by means of the familiar relation, 



COS W, =: 



+ cos2w. j /I -t- cos 2 Wo 

 I andcosw=/ - - 



But it is easy to see that these double signs, in each case, only refer to the 

 two opposite branches of the same line in respect of the origin : so that, on the 

 whole, the solutions are found to be single, and that there is one, and only one, 

 pah* of lines which fulfils the conditions of the porism. 



VOL. XV. PART IV. 7 X 



