58 Statics. 



By a course of reasoning similar to the above, we should 

 find that the distance GG", of the centre of gravity from the axis 



A p is f 

 J 



AM 



Such are the general formulas which serve to determine the 

 centre of gravity of any arc of a curve of which we have the 

 equation, by means of the lines designated by # and y. 



103. If the arc of which we wish to find the centre of gravi- 

 ty, is composed of two equal and similar parts AM, AM, situated 

 Fig. 41 * on each side respectively of the axis of the abscissas, it is evident 

 that the centre of gravity G, will be in the straight line AP ; we 

 have, therefore, only to find its distance from the point C. Now 

 it is plain that the moments of the two arcs Mm, M rri, with 

 respect to the axis NN', being equal, the distance GC will be 

 equal to 



76. 



*) 



MAM' 



For example, let the arc MAM be an arc of a circle ; we 

 Tri g- have y = Vax a; 2 , a being the diameter. We shall easily fincl. 

 Cal 98. an d indeed we have already seen, that 



adx 



We shall have, therefore, 



h x dx 



= af(h x)dx (ax x 2 )~~ *' 



Supposing now for the sake of greater simplicity, that the 

 point C is the centre, then JlC = h = a ; we shall have, there- 

 fore, 



2 f (hx) ^dx^+dy* = a/(irt a?) dx (axx 2 )"^ 

 J 



an integral to which no constant is to be added, because when 

 # = 0, this integral becomes zero ; as indeed it ought, since the 

 sum of the moments is then evidently nothing. 

 We have, therefore, finally, 

 i x) d x 



