DR MATTHEW STEWART'S GENERAL THEOREMS. 587 



Such is Dr STEWART'S statement ; but forming the equations of condition, 

 as in the preceding Propositions, we have 



3 ^ ( a m P m COS 0) = Sa m fa COS W l + - +?3 COSW 2 } ..... (2) 

 3 S ( a m P m Sln e J = fifo fol Sln W ! + - ' - + ?3 Sln W 3 } ..... ( 3 ) 



3 S(a m cos20J = Sa m . {cos 2 WL + . . . + cos2wj ..... (4) 

 3 S (a sin 2 6 ) = S a . {sin 2 w, + . . . -t- sin 2 w,} . (5) 



V TO TO' TO L O J V / 



Now, in the present case we have only five equations for the determination of six 

 quantities, w 1 , w 2 , w 3 , y^ q^ and y 3 . The condition of the porism cannot, therefore, 

 be fulfilled without another condition. 



This indeterminateness, in respect of this proposition, has been noticed by 

 Dr SMALL, Ed. Trans., ii. p. 46. In the discussion of Props. 46-53 of this Series, 

 will be noticed again. 



PROPOSITIONS XXIV., XXV. PORISMS. 



Let there be given m lines and m magnitudes as before : then p straight lines can 

 be found, such, that if we draw the perpendiculars from any point Z to all the 

 lines given, and to all the lines found, we shall have 



fr Mi ., fi ^^. p[8 ( a ' zp ~*' )}=Sa ' S( > z W- bu ........ ,H.,vr 



The general form of the component terms of this equation is, Lemma ii., 



Whence, forming the equations of condition, we have the following series, 



P S(a m cos3ej = Sa m .S(cos3&J 4 ) 

 p S (a m sin 3 6j =Sa m .S(sm3u^) 

 P S K P m cos 2 J = S a m . S (ff t cos 2 w 4 ) 

 P S ( a m P m sin 2 6 J = S a m S (9 t c s 2 W 4 ) 

 P s ( a m cos OJ = Sa m - S ( cos W 4) 



P S ( a m Sin 6 J = S(l m - S ( Sin W 4> 



P S ( a m P m " COS 6 J = Sa - S (i ^S W 4 ) 



