84 Statics. 



Let us suppose, in the first place, that each knot connects 

 only three cords, and that they are all in the same plane, as 

 represented in figure 58. 



The power p is exerted against the two cords A g, AB. Let 

 the directions of these cords be produced ; having taken AF 

 to represent the power p, we form upon AF as a diagonal, and 

 upon the prolongations AE, AD, as sides, the parallelogram 

 ADFE. The force p will be expressed by AE^ and the tension 

 of the cord BA by AD ; so that, denoting by a this tension, we 

 shall have 



p : p : a :: AF : AE : AD 



: : sin DAE : sin FAD : sin FAE 

 : : sin p AU : sin FAD : sin FAE. 



Suppose the effort AD applied at B, according to BI, in the 

 same straight line with AD, and equal to AD. The force BI is 

 exerted against the power q, and against the cord BC. By pro- 

 ducing, therefore, as above, the cords q B, CB, and forming the 

 parallelogram GBHI, BH will represent the value of the force 

 q, and BG the tension of the cord CB. We shall accordingly 

 have, b denoting this tension, 



a : q : b : : sin GBH : sin IBG : sin IBH. 



Suppose the effort BG applied at C, according to CK, in a 

 straight line with BG, and equal to BG. The force CK is exert- 

 ed against ar and against r. If therefore we produce r C, zr C, 

 and form as before the parallelogram MCLK, CM will express 

 the value that must belong to the force r, and CL that which 

 must be exerted by & ; whence, 



b : r : *r : : sin LCM : sin KCL : sin MCK. 



If we would have immediately the ratio of the tension Q of 

 any branch $A of the cord to the tension of any other branch, C ar, 

 for example, it may be readily obtained in the following manner. 



Of the series of ratios above found, if we take only those 

 which relate to the tensions of the parts of the cord, g ABCvr, we 

 shall have 



p : a : : sin FAD : sin FAE, 



a : b :: sin GBH : sin IBH, 



. 6 : *r:: sin LCM : sin MCK; 



