60 Statics. 



pect to JVC, will be (h x) y d x. Therefore, the distance GG' 



of the centre of gravity from the base, will be - ^MM ' 



Now calling c the base, we have 



AC : AP :: JVJV : MM'-, 

 that is, 



ex 



li : x : : c : y == -r- j 



c x d oc 

 therefore f(h x) y d x becomes f(h x) ^ , or 



t-L (hxdx x 2 dx\ 



it 



the value of which is (-^ -|- J or C ~ (3 h 2 a?). Now 



the surface AMM' is -^ or ^-r- ; therefore the distance 



Z li 



GG of the centre of gravity of the surface from the base is 



or 1(3 A 2x), 



which, when x = h, becomes 7i, to which this distance is there- 

 fore equal. Now if we draw the line AGL, the similar triangles 

 ACL, GG'L, give 



LG : LA :: GG' : AC : : i h : h : : 1 : 3 ; 



93 therefore LG = 1 Z*#, which agrees with what has been before 

 demonstrated. 



106. Let us now apply the formulas to curved lines. Sup- 

 pose that APM is a portion of a circle whose diameter is a, and 



Fig. 44. whose centre is C ; we have then h = AC = | . Now 



y = \/ax x 2 ; 

 the quantity /( h x) y dx, becomes therefore 



Cal. 88. f(a x)dx Vaxx 2 , 



or /(i x) dx (ax a? 2 ) *, which is an integrable quantity, 

 and being integrated, gives 1 (a x # 2 ) * ; a quantity to which 

 no constant is to be added, because it becomes zero, when #=0 ; 

 as it evidently ought. We have, therefore, 



