DR MATTHEW STEWART'S GENERAL THEOREMS. 595 



The next eight porismatic propositions (4t> to 53 inclusive) which close the 

 series of porisms respecting points and lines, are, in fact, but varieties of the same 

 general proposition, according to the positions of the given lines, the form of the 

 number n, and the relative values of the magnitudes a v a 2 , . . . , a m . 



PROPOSITIONS XLVI. TO LIII. PORISMS. 



Let there be given m lines, and as many magnitudes a^ a^ . . . a m : then there 

 can be found p other lines, such, that if from any point whatever Z, there be 

 drawn perpendiculars Z A x , Z A 3 , . . . . Z A m to the given lines, and ZBj, 

 Z B 2 ..... Z Bj, to those found, ive shall have, for values of n, subject to con- 

 ditions which may be determined 



P S(a m .ZA m n ) = Sa m . S (Z B/) 

 The perpendiculars being respectively, 



Z A m = + {p m r cos (66 m ) } 

 ZBp = +{g p r cos (0-Up~)} 



we shall form the expression by means of Lemma ii. Also, as the process is 

 general, we may, in conformity with previous practice, omit the double sign of 

 these perpendiculars. 



Taking, then, the conditional equations furnished by the developments of 

 Lemma ii., we shall have in succession : 



First, from terms clear of 6. 



r" ____ p.S(a m p m n } =Sa m .S (?/) 



r 2 ____ p.S (a mfm n - 2 ) = Sa m .S (q^} 



r* ____ p.S(a m p m "-*) =Sa m . S 0/~ 4 ) 



Second, from terms in cos Q and sin 6. 



r . .. .p . S (a m p m n ~ l cos 6 m ] = S a m . S (ft)"" 1 cos Up) 

 r 3 ____ p.S (a m p^~* cos Q m } = Sa m . S (q p n ~* cos w p ) 



and 



r ____ p.S (a m p m "~ l sin Q m } = Sa m . S (g^ 1 sin 

 r ____ p . S (a m p m n ~* sin Q^Sam.S (ft,"" 8 sin 



