108 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[ApilIL, 



-— ,- , -^ &c. = 0, since by our hypothesis the velocities are 

 zero, we have : 





+ y 



df' 



= 



(1) 



(*-^n)5> + (C-.V„)5^ =0 (u + 1) 



Now, the second differentials express the accellerations of the 

 several bodies parallel to the axes of .v and y. Also the forces 

 acting on each body vertically downwards are the difference of the 

 resolved vertical parts of the two tensions acting on it, and the 

 force of gravity. The horizontal force on each body is the differ- 

 ence of the resolved horizontal parts of such tensions. 



Let T,, T,, T, T„+, be the tensions of the 1st, 2nd, 3rd 



(» + l)th side of the polygon respectively. 



The resolved vertical parts of these tensions are 



T,^ 



~ 1 'T' "^ "^ 



T„+i — — respectively. 



a - a 



The resolved horizontal parts of these tensions are 

 T, yi — Vi -p Pa—y 



T„+i- — — respectively. 

 a 



a - a 



The equations of vertical motion are (m being the mass of each 

 body, and g the force of gravity) 



d-jc„ 



.,-T.^ 



mg—T^ 



dt- 



-fT 



df- 

 &c. 



■^2 I "T" *^ i "^3 



&c. 



dt- a a 



and the equations of horizontal motion are 



Jjli rj, y^-Vi 



dt' 



= T,+ 



c— J/„ 



- T, 



y^—yi 



■■ a 



ys—yi 



3/n — y—i 



(3)' 



(1)" 



(2)" 

 (3)" 



We have then, in all, {3n-\-\) equations, involving 3re+l un- 



d-x 

 known quantities — namely, the n differentials -—,, the Ji differen- 



df-' 



d-y 



tials— ■/, and the n+1 tensions T. Our object is now to eliminate 



the differentials, and so obtain n-\-l equations involving no other 

 unknown quantities than the T's. 



If »,, H.;, e., e„+i be the angles at which the 71+1 sides of the 



polygon are respectively inclined to the vertical, it is easy to see 



&c, 



b—x, 



cos fl.+i, — — = sin 8„+i. Substituting these values 

 a a 



in the 3 n-\-\ equations, we shall obtain, after some trigonometri- 

 cal operations which are here omitted for the sake of brevity, the 

 following symmetrical results: — 



mg cos e^ — Tj-fTj cos {,8.,— 6^) = 0' 



Ti cos (^2— fJ,)— 2TJ+T3 cos (e^—eS = o 

 'i\ COS (^3— e^)— 2T,4-Tj, cos {e^—eS = o 



T, cos (d,-e3)-2T.,+T, cos (e^-ej = ^- («) 

 &c. 



T„_,cos(5„-e„_,)-2T„-i-T„+,cos(^„+,-ft,) = 



T, cos(0„+i— 5„)— T„+i— iH^cos (9„+i = 



The first and last of these equations might have been obtained 

 independently, by applying with respect to the first and the last of 

 the moveable bodies the consideration tfiat wlien a body begins to 

 move, the sum of the forces resolved at right angles to the initial 

 direction of motion must be zero. 



It is theoretically possible to effect these solutions of (n+1) 

 equations of the first degree, involving (k+1) unknown quanti- 

 ties. The law which the values of the unknown (piautities of a 

 general system of equations of the first degree follow, was first 

 observed by Ci-amer, and subse<|uently expressed in a somewhat 

 more convenient form by Bezout in his Tlnorie Generate des Equa- 

 tions. The first rigorous demonstration of the rules was given by 

 La])lace in the Memoirs of the Academie des Sciences for 1772 (2rfe 

 pnrtie, p. 294). A very elegant demonstration is also given by 

 M. Gergonne in his Annates, vol. iv. p. 148; and a subsequent paper 

 by him on the same subject is to be found in vol. xii. p. 281, of the 

 same periodical. These investigations do not, however, present 

 the required values in the form of general expressions, but merely 

 furnish rules for constructing such an expression by writing down 

 in a system of («-|-l) equations, the 1.2. 3...?j-(- 1 permutations of 

 ()i-f 1) quantities. 



It is, therefore, necessary to adopt an independent method in 

 the present instance, in order to determine the unknown quanti- 

 ties in the series (a) of equations just given. The solution re- 

 quires considerable care in order to avoid an excessive complexity 

 of the results. 



Let icos(9j-e,) = «oi Jcos(e3-fl2) = M,; ^ cos (9,-83) = Uj; &=• 

 ^ cos(fl„+i — 6„) = M„_i (A) 



Then omitting for the present the first and last of the equations 

 (a) just given, the remainder may be put in the following form : — 



1 M„ 



1 „ M, _ 1 — ap «„ 



T. = T3 T^-i- = T^ i-T,— 2- 



T, = T^ T.— = T„ ! — ^-T,-^ ^ 



T = T --T "^ = T l-"?-"g-«g + "?"g _ T.«o(l-«i-»D 



<= =«^ ■*«< "" 2 «i"2«3«4 »l"2"a«4 



&C. 



T„^,= T„-i-_T„_,"r: = T, ^ T.«„ ^ 



The second values of T^, T^, and T^, above given, are ob- 

 tained by substitution of the previously-ascertained values of T3, 

 T|, and T,,, respectively. Our object is now to ascertain the law 

 of the co-efficients of T^ and Tj on the right-hand side of tlie 

 above equations — that is, to find general expressions for a and b- 



Let «? be written = 1'; m5=2'; M|=3'; &c. «^^=j»'. Then 

 we sliall find, by continuing the process of substitution, that 

 for T„ a = l-(l'-f2'-t-3'-1-4)-|-l'(3'-f4')+2'4.' 

 forT,, a=l-(l'+2'4-..5')-(-l'(3'-t-...S')-f2(4'-f5') + 3'5-l'3'5' 



for T„ a = l-(l'+2'-|-...6')-t-l'(3'-l-...6')-(-2'(4'-h-6')+3'(5'+ti') 

 -I- 4'6'— 1'3'(5'-|-6')— 2'4'6'. 



The law observed by the above successive values of a is tolerably 

 obvious; and from the manner in which they are derived, is evi- 

 dently continuous. 



In order to indicate the law by a general expression. 



Let r4-2'-|-3'-t-...m' — S(m) the sian of the quantities I', 2", 3', &c. 



Let r(3'-l-4'4-...m')-t-2'(4'+5'-|-...w')-|-3'(5'-|-...m')-f 



-\- {in—'-Z)'m' = S (?»— 2, m) the sum of the products of every two of 

 them omitting those products in uhich any one of the quantities is mul- 

 tiplied by tlie next in numeral order. 



Let l'|3'(&' + (i' + 7' m') + 4'(f>' + 7' + . ...m') + 5'C7'+ »i') + +(m-2/m'} 



+ 2'{4'(6'+7'+8' mO + S't7' + t' + m'J + 6'(8'+ m'j + +{m-2///i'} 



+ 3'{5\7'+8'+9'.. ..>«'; + 8\»' + a' + m') + 7'vS(' + m', + +(ni-2/m'} 



+ &e. + ',m-4)' (m-2)' m' 



= S(m — 4 , m - 2, m) Me sum of the products of every three with like omission. 

 Similarly, let the series of which the last term is 



(m— 6)'(m— 4)'()"— 2)W = S(m— 6, m— 4, m— 2, m) 

 and that of which the last term is 

 (»j — 8)'('"— 6)'(m — i)'{m—2)'nt = S(m— 8,m— 6,m— 4,ni— 2,m). 



