P R I S M S. 



113 



Forum*. 



4 ; ~ 



+ 



^) 



~ 



2- 1 ) 



but the value of the series sin. 0* -f- sin. -f- 



sm. + 



2 



is consequently the sum of the 



squares of all the chords is equal to 

 t 



which is a constant quantity. This produces another 

 jiorism, which comprehends the last as a particular 



case. 



A circle being given, a point within it may be found, such 

 that, if ami number n of chord* be drawn through it, 

 making equal ung'es ivilh each other, the sums of the 

 squares of these chords shall be equal to the square descri- 

 bed in a given line. 



A circle being given, a point within it may be found, 

 such that, if any number n of chords be drawn through 

 it, making equal angles with each other, the stim of the 

 fourth poivers of those chords shall be equal to a given 

 fourth power. 



The fourth powers of these chords are thus algebrai- 

 cally expressed, 



2 4 (r 4 2 r 2 t>* sin. 0* + t> 4 sin. 4 



( 

 r 4 _ 



2 r 2 t>* sin. 



Sln - -- 



" 2 2 4 \ 



.t> 4 sin. 0-4 a- ) 



n 



the sums of series containing the powers of the sines 

 or cosines of arcs in arithmetical progression have long 

 been known,* and when the common difference is the 

 twentieth part of the circumference, and the number 

 of terms equal to n, they receive a very remarkable 

 simplification, for in that case 



And also, 

 Cos. 2m 4- cos. 04. 



. . COS. 0-4 if = 



n 



2m. 2m 1 . ..m 



*sin.0 



And if these expressions are expanded, the eri* of the 

 powers of sine*, which constitute each vertical column, 

 are each equal to some constant quantity : the whole 

 sum is therefore independent -of '. 



A straight line and a circle being given, and also a 

 point in that diameter of the circle which is perpendi- 

 cular to the given Hue, it has been found that the sum 

 of the perpendiculars to the given line drawn from the 

 extremities of any chord passing through the given 

 point, is expressed by ' 



2 (a v -f v cos f) 



If another chord pass through the same point at right 

 angles to the former, the expression for the sum of ihe 

 perpendiculars drawn from its extremities to the given 

 line, will be 



2 (a v -f. wbin. #) 



The sum of the four perpendiculars is therefore 

 4(n t,)^. 2t; = 4a 2. 



This produces the following porism : 



A circle and a straight line being given, a point may 

 be found nithin the circle, such that, if any tno chord* 

 be drawn through it at right ar.gles to each <>lher, and if 

 from the extremities of these chords perpendiculars be 

 drawn to ihe given tine, ihe sum of these four perpendi- 

 culars shall be equal to a given right line. 



This property may be generalized, by supposing n 

 chords, instead of two, to be drawn through the point 

 found, and it will be perceived, that, if they make equal 

 angles round that point, the sums of the perpendiculars 

 drawn from their extremities to the given line will be 

 a constant quantity. 



In the same figure, the sums of the rectangles under 

 the perpendiculars, let fall from each chord, will be 

 thus expressed : 



(a v}- 4- (2 at; w 2 r 2 ) cos. 





1.2. m 2 2m 



where m must be less than n ; if we apply these con- 

 siderations to the several series in the above sum. which 

 respectively multiply 2 r 2 t> and t; 4 , we shall find for 



the value of the first, and for that which multiplies 



S 



v > T~ n > so " lat tne sura t all the fourth powers of 



(a i-) 2 + (2 a v t- 2 ; 5 ) cos. 

 And since 



* , ~ *~ , 2 M 2 

 cos. <" 4- cos. 6 -\ 1- . . cos. t + - 



N H 



we have for the sum of all the rectangles 



the chords is 



2 4 n r 4 2 4 n r 1 1 -f 6 n t 4 . 



If m rs less than n, then it maybe readily shown 

 that if chords be drawn through IDY point within a 

 circle, making equal angles with rach other, the sum 

 of the 2 m powers of those chords will always be equal 

 to a constant quantity ; for the 2 m powers of such 

 chords are 



2 a v ;* r~ 



n. 



This suggests another porism, as follows : 

 A straight line and a circle tiring gin /;, a f vint nithin 

 may be found, such that a certain number n of chords 

 being drnmn through it ; ihe sums of aU the rtctagle* 

 under the perpendiculars, dran-n f'rcm the extremities of 

 each chord, shall be equal to a gii-eK square. 



VOL. XVII. PART I. 



Acad. Pttrop. Ccm. JVtv. 1773, Euler. 



