Rope Machine. 85 



these being multiplied in order, we have 



g : v : : sin FAD sin GBH sin LCM : sin FAE sin IBH sin MCK. 



If we would have the ratio of the tension g, to the tension 6, 

 we should multiply only the two first proportions. The other . - 

 ratios may be found in a similar manner. 



If it were proposed to determine the ratio of the powers 

 among themselves, we have only to deduce from the above series 

 of ratios the ratio of two consecutive powers to the tension of 

 the same cord ; thus, 



p : a : : sin g AD : sin FAE 

 a : q : : sin GBH : sin IBG 

 q : b : : sin IBG : sin IBH 

 b : r : : sin LCM : sin KCL. 



Taking the product of the corresponding terms, and reducing, 

 we have 



p : r:: sing AD sin GBH sin LCM : sin FAE sin IBH sin KCL. 



To obtain the ratio of p to q, we should multiply only the terms 

 of the two first proportions. 



It will hence be seen how we ought to proceed when there is 

 a greater number of powers, or when we would compare the 

 tensions of the cords with the powers themselves. 



149. If the powers p, q, r, bisect the angles g AB, ABC, BC &, 

 respectively, the angles DAF, FAE, would be equal ; and the 

 angles GBH, LCM, would have the same sines as the angles 

 IBH,MCK, respectively*; whence, by means of the above ratios, 

 it will be seen that the different parts of the cord g ABC & would 

 be equally stretched. 



1 50. If instead of the powers p, q, r, we substitute in A, B, C, 

 fixed points or pivots, the pressure upon these points arising 

 from the tension of the extreme parts of the cord, would be 

 directed in such a manner as to bisect each of the angles ; and 147. 



the tension of the several parts of the cords gA, AB, &c., would 

 be equal. Accordingly, if two powers g, w, are exerted upon a 

 cord passing round the periphery of a polygon or of any curve, Fl *8- 59> 

 the tension will communicate itself equally to every part, so that 

 the two powers must be equal. 

 * * &3Hsz / HMCss sin 13H', V *i*L CM~ // M C1*~ t 



