112 



P O R I S M S. 



IWisms. 







ccci xvn 



Fig. 7. 



8. 



2 (a a-f t;cos. 4 s )-f-2 e=2n (a i/ -ft/ cos. o 1 ) 

 or, 1 a n t/-f v c-f (nv' v) cos. 2 =0. 



If we assume v nv', and n la c = 0, this 

 equation is verified without assigning any particular 

 value to cos. e ; hence result various porisms by giving 

 particular values to n ; if n = 2, we have the follow- 

 ing: 



A line LM (Fig. 7- ) and two points C and C, in a perpen- 

 dicular to it being given, a third point may be found about 

 which if a circle be described with any radius OR, and if 

 through the two given points any two parallel chords PQ 

 and P / Q / be drawn ; the sum of the perpendiculars PL -f- 

 QM drawnfrom the extremities of one of these chords, to- 

 gether with a line which may be found shall be equal to 

 twice the sum of the perpendiculars PL+QM drawnfrom 

 the extremities of the second chord to the given line. 



The demonstration is sufficiently obvious from the 

 algebraic investigation. The centre of the circle is si- 

 tuated in the same line with the two given points, and 

 its distance OC from the point C is equal to CC, and 

 the line which may be found is equal to OF. 



This porism is rather remarkable from the circum- 

 stance of two of the quantities being indeterminate, 

 namely, the angle which the chords make with a given 

 line, also the magnitude of the radius of the circle ; 

 and it may be observed that instances of such double 

 or even triple indeterminations will occur much more 

 frequently in the algebraic discovery of porisms than 

 in their geometrical invention. 



If in the problem from which this porism was de- 

 duced, we had supposed the chords at right angles to 

 each other instead of being parallel, then we should 

 have had the equation 



2( v+v cos. 0*-f c)=2?z(a t/-ft>' sin. o*) = 



2? (a v' cos. 4 2 ) 

 Hence, 



n 1 a -ft; -c 



cos. t* =. 0. 



This equation may be satisfied without determining 

 the value of e, by assuming the two equations, 



n 1 a-f-v c=0 

 and nv'-f-r=0. 

 From the latter we have 



And from the former 



v rr c n 



1 a 



This produces the following porism : 



A point O and a right line LM (Fig. 8.) being given in 

 position, two other points C and C may be found suck that 

 if a. circle be described round the first point O luilh any 

 radius, and if two chords PQ and I^Q, be drawn at right 

 angles to each other through the latter points the sum of 

 the perpendiculars PL-J-QM drawn from the extremities 

 of one of those chords to the given line, together with a 

 line which may be found, shall be equal, to a given multiple 

 n of the sum of the perpendiculars P / L / -j- Q,M, drawn 

 from the extremities of the other chord to the given line. 



The radius vector from any point within the circle 

 bein 



CP = v cos. 



v' 2 sin. f- 



That which is distant from it half a revolution will 

 be, 



And consequently the expression for any chord passing PoriB 

 through a point C distant from the centre by the quan- 



tity v, will be 



PQ = 2 



v 1 sn. 



and the value of any other chord parallel to this at a 

 distance from the centre, denoted by v', measured on 

 the same diameter, is 



P,Q, = 2 V r v'z sin. P 



Let us now suppose three chords parallel to each 

 other, and that the sum of the squares of the first two 

 is equal to twice the square of the third ; the equation 

 which results is 



4(r 2 v 2 sin. 4)-f 4(r 2 u' 2 sin. ^)=8(r " sin. 

 or 2 r* (t> 2 + v' 2 ) sin. t = 2 r 2 2 t>" 2 sin. ft 

 and this can only be satisfied by supposing 



if the origin of the co-ordinates, which is now at the 

 centre, be removed the distance a, then v, v', and v" be- 

 come a-\-v, a -ft/, anda-fw"; and the last equation 

 gives 



if v, v', and v" are given, a may be found, and will 

 be 



a = - ; j-/> 



This gives rise to the following porism : 



Three points C, C /} and C% , being given (Fig. p.) PIATE 

 in a right line, another point O may be found such that J/"CCCCLXV 

 with any radius a circle be described about C as a centre, ^'S- p< 

 and if three parallel chords be drawn through the three 

 points, then the sum of the squares of two of them, PQ 2 

 and P,Q y 2 shall always be equal to twice the square of the 

 remaining chords P Q 2> 



It has been shown that if any chord be drawn through 

 a point C, we have 



PQ 2 = 4 r 2 4 v~ sin. f- 



If we consider another chord at right angles to the 

 former, and drawn through the same point, we have 



P,Q, 2 = 4? r~ 4? v 2 cos. # 

 the sum of these two is 



PQ 4 + P.Q/ 2 = 8 r 2 4v* 



which is a constant quantity ; and since the angle ( is 

 variable, this suggests the following porism : 



A circle PPQQ (Fig. 10.) being given, another circle Fig. l 

 may be found, such that, if through any point of its cir- 

 cumference two chords be drawn to the Jirst circle, and 

 perpendicular to each other, the sum of the squares of these 

 two chords shall be equal to a given square. 



If through any point in a circle n chords be drawn, 



making with each other the angle these squares 

 will be represented by 



t; J sin. <H -- 



4-P 

 r t)*sin. *-| -- 



:=v cos. 



sn. 



