Om Chain Bridges. 373 



greater than that at o, and greatest of all at the points of siispen- 

 sion.-^Hence we learn, that to make the strength of the chain 

 in proportion to the stress, which reason and science require, its 

 dimensions must vary from the lowest point o to P. This in- 

 crease in size and weight will change the curve from a catena- 

 rian to one of a more complex and transcendental nature, wliich 

 m>ue but the most transcendent abilities will be able to develop. 

 We therefore mpst content ourselves vvilh the approximate hnow- 

 ledge derived from considering the chain of invariable dimensions, 

 which is a problem of difficulty and labour, as is well known to 

 inathematicians ; and none other need attempt it. I shall not 

 give tlie elementary investigation by which the following equa- 

 tions and a knowledge of the sul)ject can be derived ; it may be 

 seen in the authors before quoted, some of v.'ht)m have given ex- 

 amples of numerical calculation, which may be useful to those 

 who have not laboured much in the difficult parts of science, but 

 may be desir(nis of ascertaining the results of the pre'sent or 

 some future project of this nature, putting a to denote the stress 

 or strain at the lowest point o; Go = a-; GQ=y ; and .Qo=a;. 

 The following aie the general equations alluded to : 



1 ^ 11 a-\-x-\-\/'lai-\-x'- 1 , ^+ V "-+':' - 



1st, ?/ = a xhyp.log. -!— ^ ■ — =axhyp.Iog, - — — , 



2d, «= 'v/2«x+ x- 



3d, x= —a+ -v/ a- + x": from which, if any two of the quanti- 

 ties be given, the rest may be found. But it is evident, from the 

 transcendental nature of the curve, that this can only be done in 

 the present form, bv the tedious method of trial and error, a mode 

 of operation not suited to common calculators. To obviate this 

 difficulty in some degree, Dr. Hutton, at p. 33 of his Principles 

 qf Bridges, has reduced the first equation into the following series : 



a=ixxfi + l-^+ 3*^ - ,|^\, ^-c. « Where a 



few ternis," he says, "are sufficient to determine the value of a 

 pretty nearly." Perhaps the above series will be better adapted 



to calculations, bv substituting v=. — and reducing the fractional 



coefficients of the series into decimals. Thus a=\yv — ' x : 11* + 

 ; — •1777y~^ + "lH52v— ■» — •05l37v — ', &c. a being found, 

 the length of the chain and other particulars may be obtained 

 by substituting the value of a and x in the second general equa- 

 tion. If the length of the chain and distance of the points of 

 suspension be given, the value of a may be found, by trial and 

 prror, or in a series in terms of y and x by a similar process tp 

 jhat pursued by Dr. Hutton. 



A a 3 Having 



