DR MATTHEW STEWART'S GENERAL THEOREMS. 593 



expansion, to zero ; and we shall thus obtain the conditional equations of the 

 porism, viz. : 



First, from terms clear of 6. 



... . P S(a m r- <"-i>) = ,sr w . 

 -Sa r^~ = Sa . S 



Secondly, from terms in cos Q and sin 6. 

 r....pS(a m r m * n ~i cos 0J = Sa m . 5 (V "~ X cos w p 

 r 3 pS(a m r m 2 "- 3 cos 6J = S a m . S ( p s - cos W|> 

 ^ pS(a m r m 2 "- 5 cos 6j = S a m . S (/- cos Wp 



r 2 "-' -P8(a m r m cos 6j =S a m . S(u pCOS U p} 



and 



pS(a m r m sin d m ) =Sa m . S(u pS in^ 



Thirdly, from terms in cos 2 6 and sin 2 6. 

 . P S(a m rj n ~ 2 cos 2 O m ) = Sa m . S (V- 2 cos 2 



~* cos 2 



r s ~ 2 . JB S ( m r m ^ cos 2 6 m ] =Sa m .S (u p * cos 2 UP ) 



and 



r* v .. . p S(a m r m 2 "- 2 sin 2 6j = Sa m . S (u p *-* S in2u p 

 r* . . . . p S (a m r m *-* sin 2 6>J = S m . >S (;>-* sin 2 Wp 



r 2 "- 2 . p S (a. r m 2 sin 2 0J =^a m . >S (V sin 2co p ) 



Proceeding in this way, we arrive at, 



Lastly, from terms in cos n 6 and sin n 6. 

 r"....pS(a m r m n cos n m ) =Sa m . S (u p n cos w p ) 

 r" . . . . p S (a m r m n sin n m ) =S m . -S ( p n sin n 0), ) 



VOL. XV. PART IV. 7 X 



