THE EQUILIBRIUM OF BODIES IN CONTACT. 489 



"ly fi^ dy TTe ^ e * 



sin--cot--cos^gtan| 



2{co4sin=|-sin^|eo4)- 



Whence, by Lagrange's Theorem, neglecting powers of a above the 

 hrst, we have, by equation (60), 



P= \X+ar\^,-Q)} coti'o 



(cosS^„-cos9)^(sin^gcot|Vcos-|tan|)(^„-9)-gcos9cot>P „) 



—/«/•■, 



P= |X+ ar^C^P^- e)} cot I', 



_ i(l-tan-|-°)(l-tan-|)g-(tan-|Vtan-|)(^„, e)] 

 (tan^l^.tan-'Dsin^J-^ 



iar'. 



P^Xcoti- 





Mtan^f-tan=^)tan|-» 



Equations (62), (63), {Q5) determine the vahies of P and ^P that is 

 the pressure upon the key-stone and the positions of the points of rupture' 

 for^every condition of loading, and every form of the gothic and circular 



Assuming h = o and eliminating the value of tan^i, we have, by 

 equations (62), (63), and (6.5), 



