582 MR THOMAS STEPHENS DAVIES ON 



and 



sin 6 l + sin 2 + sin B 3 +.... + sin 6 n = 



sin 2 0j + sin 2 6 2 + sin 2 3 +.... + sin 2 B =0 

 sin 3 0j + sin 3 2 + sin 3 3 + .... + sin 3 B =0 



* 



sin (-l) Q! + sin ( 1) 2 +....+ sin (w 1) n = 0. 



These properties are already known, and hence only require to be put down, 

 without investigation. 



SECTION II. THE PORISMATIC PROPOSITIONS. 



PROPOSITIONS IX., X. PORISMS. 



Let there be given in a plane the m points A t , A 2 , . . . . A m , and as many magni- 

 tudes p a 2 , . . . . a m : then a point X may le found, such, that if we draw 

 AI X, A 2 X, . . . . A m X, and likewise to any point Z in the same plane we draw 

 A t Z, A 2 Z, . . . . A Z, and join X Z, we shall always have 

 a t . A,Z 2 + a 2 . A 2 Z 2 + ---- = 0l . AX 2 + a 2 . A 2 X 2 + ... -^(a^a^ + . ..)XZ 2 : 



that is, 

 S(a m .A m Z-) = S(a m .A m X?) + Sa m . XZ 2 . 



For, let the given points be denoted by i\ n r 2 6 2 . . . r m Q m ; the porismatic 

 one by r 6 W and the arbitrary one by r &. Then the general type of the compo- 

 nent parts of the equation of the porism are 



a m . A m Z 3 = a m {r* -2 / r m cos (5 - ffi ) + r m *}, 



a n . A m X 2 = a m {r 2 -2r - m cos(0 - 6^ + r^}, 



Sa m . XZ 2 = Sa m {r* ~2rr cos (6 -0 ) + r 2 }. 



Inserting these in the general equation, cancelling common terms from the 

 equation, and equating to zero, the co-efficients of r cos 6, and r sin 6 (the only 

 forms in which the arbitraries appear in the expression), we shall have the follow- 

 ing conditional equations : 



S a m . r cos 6 = S (a m r m cos 6 m ) ........ (1) 



S a m . r c sin = S (a m r m $m.Q m } ........ (2) 



S a m . r a 2 = r<, cos 6 a . S (a m r m cos 6 m ~) 

 + r sin # . S (a m r m cos 6 m ~) 

 The first and second are the equations of the centroid, and the third is in- 

 volved in the other two, as is obvious. Wherefore the porismatic point is the 

 centroid of the system. 



[See, also, the note on these propositions.] 



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