1819.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



109 



Then, for the general value of a in the equation for T„+], where 

 m=n— 2, it will be found that 

 a = 1 — (Sn)— 2 + S(n— 4., «— 2) — S(n— 6,7»— I, n-2) + 



7!-2 



( — ) "' «2?(4?/g «„_o when n is evenl 



( — ) 2 tij^ii^v^ u„_2 when )( is odd \ 



In the same way b may te determined ; and it will be found 

 that the expression for b is identical with that for a, except in 

 that the quantity 1', and all terms multiplied by 1', are to be omit- 

 ted. 



If we suppose the quantities 1', 2", 3', &c. to follow a general law; 

 that is, if the inclination of the sides of the polygon be expressed 

 by a general formula dependent on the order of tlieir succession, 

 the series for a and b involve Finite Diiferences, and if the ope- 

 rations be not too complicated may be summed by the calculus of 

 Finite Differences. 



We will take the most simple case — that where the polygon is 

 inscribed in the arc of a circle. Here by the geometry, 8^—6^ = 

 e^g— flj = 6.1 — 63 &c., or r, 2', 3', &c. are all equal (=»-, suppose). 



In this case, by the meaning assigned to S(m) u--\-u--\-u- 



(m times) = S(m)^M-»i, 



^ ^ (m-2)(m-l) 

 S (»i— 2, »i) becomes (m — 2 + m — 3 + m — 4 + .... 1)m* = u" 



Or reversing the order of the terms, we have series of the re- 

 spective forms 

 S(»J-2,»i) = «''(l + 2 + 3 + 4 + 5 + ..m-2) = u=2(m-l), 



S(»i-4,m-2,»i) = u6{l + Cl + 2) + (l + 2 + 3) + .. +(l + 2 + 3 + ..i»-4;} 

 = u«2^(m-3), 



S(m-6,»)-4,m-2,m) = us{(l + 1 + 2) + (1 + 1 + 2+ 1 + 2 + 3) + 



+ (1 + 1 + 2 + 1 + 2 + 3 m)} = «92»(m-5), 



&c. 

 where 2 is the symbol of integration in the Calculus of Finite 

 Differences. The numbers expressed by these series are technic- 

 ally known as the figurate numbers. 



Now by the principles of the Calculus 



(«-l).m (m-2) {m-l)m_ Jm~Z){m-2){m-V}m 

 Sm- ^ ,2 m- 2^ ■*'» 27f.4 *''■ 



2.0.4 



Substituting these values in the series for g^, and putting 

 ra ^ « — 2, we have 



^ J ^ 2 2.3 



(n-20)(n-9)(«-8)(n-7) 

 "^ 2.3.4 ~^''- 



In the same way the value of b may be determined. 



a ^ b 



The equation T„+i = Tj 



- Ti«o: 



bined with the first and last of the series (a) of equations, suffices 

 to determine the tensions in terms of the known quantities. 



Vibrating Catenary. 



We have hitherto considered the chain as a funicular polygon 

 with the weights arranged at finite distances. When the chain 

 assumes the form of a continuous curve with the weight uniformlv 

 distributed along its whole length, the method of Finite Differ- 

 ences is replaced by the Differential Metliod, which, as is usually 

 the case in investigations of this kind, simplifies the results in a 

 most remarkable manner. 



It has been shown in the preceding investigation, that where 

 iT, y and x\ t/' are two adjacent points of the polygon, the geome- 

 trical connection of the system furnishes the relation 

 (^'-A(d'v' <Fx\ (y^-y\(d-y' d^y\ _ 

 V a /\df dfJ^y a l\dt^~dfl ~ 

 Now, when the two points in question are indefinitely near each 

 other, a—ds, where ds is an element of curve assumed at the time 



t, and x^—x and y' — y are replaced by the corresponding differen- 

 tials dx and dy respectively. So that 



••^'-■J _ /dx\ . y'-y _ /dy\ 

 a \ds / ' a \ds' 



The p.irentheses indicate that the differential co-efficients are par- 



d-x 

 tial, the time t not being supposed to vary. Also, if ^7 be a func- 



tion of s, -j^ is the same function of s-\-ds. Or expanding by 



Taylor's theorem, and, in the limit, neglecting all the terms of 

 the expansion subsequent to the second 



d'x' d-x / d\ d'-x, d\v 



\ds/ ' dt- ^ ~ did? ■ 



d'y,,,, _ d\y 



df df- 



dsdt^ 



Similarly, ^t -^^^ ^ (^) . 'l!^Jy = 



•" d<2 df- W/ df- 



So that the above geometrical relation becomes 



/dx\d^,ldy\d^ _ 



Us/dsdt'' \dii/dsd(' ^^• 



To proceed now to the mechanical equations of the problem, let 

 T be the tangential tension at the point x, y. The vertically and 

 horizontally resolved parts of this tension are respectively 



since 1^-1 and ( " J are the cosine and sine respectively of the 



inclination of T to the vertical. Now, tI-t) andTU^j depend 



for their values on the place in the curve where they act — that is, 

 are functions of s, the distance from the origin measured along 

 the curve; consequently, at the adjacent point s-{-ds, 



T (j-J and T(-t j become, expanding by Taylor's Theorem, 



and the differences between these and the former values are, in 

 the limit, 



Let lurf.v be the mass of the element ds. Then the acceleration 

 of that element vertically is the weight -\- the difference of the 

 vertical tensions, and the horizontal acceleration is the difference 

 of the horizontal tensions. \Vhence the following mechanical 

 equations : — 



or performing the differentiation of the quantity in the brackets, 

 and omitting the common factor ds, 



Differentiating these equations with respect to s, we have 



''dsdt^ VrfsV"'" \ds/\ds-/ '^ \ds-/ ' \d~s/ 



d^y_ ^ ^, / dj;y\ ,^(dT\. d^y\ ydrT \ ( dy\ 

 dsdt- \ds'y \dsy\ds-/ ^ \ds-/ \ds/' 



Multiplying these equations by \-r) ^°"l(^ -■^, respectively, and 

 adding the results, we have from (A) (rememberingthat — ^ j.- „= 1 



ds'^ds" 



and that the differential of this equation, or - „• _(-— f ^ = ( 



- ds^dir ds 



"), 



d=T 



j^ 1 (d^ dx j^ ^y dy\ _ Q 

 ^di" ds flff" ds/ 



The co-efficient of T in the bracket is equal to -., where )• is tlie 



r- 



radius of curvature. Hence the expression assumes the following 

 very simple form : — 



