598 MR THOMAS STEPHENS DAVIES ON 



First, let the given lines be all parallel. 

 The equations in this case reduce to 



p . S (a m p m * n ~ l ) = Sa m . S ( 



p S (a m Pn ? n - 2 ) = Sa m .S (?/ n ~ 2 ) 



p . S a m p m ) = S a m . S (j-p) 



which are 2 n in number, determining p=2 n, and all the lines found, parallel to 

 the given ones. 



Second, let the given lines meet in a point. 



In this case /> 1 =j2= . . . . p,= 0, and we have only those terms left which 

 involve the absolute terms of the co-efficients of the multiple cosines. Whence 

 the equations become 



p . S (a m cos 2 n 6m) =S a m . S (cos 2nto p ) 



p . S (a m cos 2 (n-1) 6 m ) = S a m . S (cos 2 (-l) w p ) 



p.S(a m cos 2 (n-2) 6 m } = S a m . S (cos 2 (n-2) 



p . S (dm COS 2 0m) = S d m . S (COS 2 Up ) 



and 



p . S (am sin 2n6 m ~) = S a m . S (sin 2 n w p ) 



p . S (a ro sin 2 (w-1) ro ) =S m . -S 1 (sin 2 (n-1) 

 jt? . S (a m sin 2 (n-2) 6 m )=Sa m .S (sin 2 (w-2) 



*..* 



p . S (a m sin 2 m ) =S a m . S (sin 2 Wp) 



Thus again, we have in this case p=2 n, all the porismatic lines passing 

 through the same point as the given ones. 



The two conclusions in the latter particular cases agree with those which 

 were found in the still more special case of Props. 15 and 19. For then n=l ; 

 and either Dr STEWART'S general form, or the one here deduced, as applied to 

 that case, is precisely the same. In his view, 1 + 1=2: in what is here found, 

 2*1=2. 



