423 The E,ev. S. Haugliton on the Thickness 



determine the value which the left-liand member assumes when 

 for a we put a + h, h being a very small quantity. 



then the equation (A) written at full lengthy after substituting 

 the pro})er values for y, m, v, will be 



Substitute a + h for a, and we have 



= 'F(a + h, f[{» + h), \ (p^xdx, \ ■\lr^xdx+\ ■\lr^xdxj. 



This admits of being exi)anded in the ordinary way. Assuming, 

 theUj for the sake of brevity, 



e = l udx, «'= I vda^j 



Jo J« 

 we have 



„ -^ , fdY dY .,, , dY , , ■ dF , , .'X ,,,, 



being the same result as would be obtained if the functions 

 retained the same form throughout. 

 "2. Let ^^^, 



The same reasoning ap[)lies. 

 -3. Let „^j. 



Substituting, as before, a + h for a, /* being positive, and writing 

 the equation at full length, we have 



Q=Y\a-\-h,fc^{a-\-h)\ (^^{x)dx-\- \ (j>c^(.v)dx, I ■\lr^{x)dx[ 



'^ Jo Jo J a+h J 



or, expanding as before. 



Again, if we put a — h for «, we find 



0=Fs a — h,f {a — h), I <p^{x)dx, I i/r,,rc?a;+ 1 -xfrc^dx > , 



'-- Jo Ja-/j t'rt J 



or 



_ „ , fdF d¥ .,, , r/F , , , ^F , , ,"1 



This result differs from (C) only in the substitution of /j, 0,, -v/r, 

 for/2, ^2J '^2> ^^'^^ ^'^*^'i agree with the result which would be 

 obtained by supposing the functions continuous. I conclude, 

 therefore, — 



