16 Mb HOPKINS, ON RESEARCHES IN PHYSICAL GEOLOGY. 



From the inspection of these equations it is manifest that 2=--; for 



putting this vahie for x. and therefore also — - for x, the two equa- 

 tions will only be converted into each other. Substituting in the first 

 equation, we have 



which gives 



1 + n(c — s%) — = 0, 



fxCS 



orcot>-2. ^-^"."":y .cot>/.-l=0 (1). 



4. Let v//|. >//,. be the two values of \|/ given by this equation. Then, 

 since the last term is -1, 



cot x//, cot \^.j = - 1 ; 

 which shews that the difference between \|/, and ^., is — , or 



^. = 2 + ^.; 



and if 0,, 9., be the values of 9 corresponding respectively to >//, and >|/2, 

 we have 



cot 6, = - 



and &, = \//, = \//,, + - . 

 The angle BPR is consequently a right angle. 



5. Since PR coincides, as shewn above, with the direction of the 

 forces of tension acting on the element' ix of the line AB, the ex- 

 pression (A) is the value of that resultant. Consequently x//, and x//,, 

 which correspond to the maximum and minimum values of the quan- 

 tity (A), determine the position of the line AB in whicii the resultant 

 of the above-mentioned forces of tension is a maximum or minimum. 



