APPLICATION OF THE PRECEDING RESULTS, &C. 43 



and consequently for that within the substance of the glass 

 itself; we shall have (art. 4) 



F=/3 and F = #'. 



One horizontal line over any quantity indicating that it belongs 

 to the inner surface A, and two showing that it belongs to the 

 outer one B. 



At any point of the surface A, suppose a normal to it to be 

 drawn, and let this be the axes of w : then w', w", being two 

 other rectangular axes, which are necessarily in the plane tan- 

 gent to A at this point; V may be considered as a function 

 of w, w and w", and we shall have by TAYLOR'S theorem, since 

 w' = and w" = at the axis of w along which 6 is measured, 



dw 1 dw z 1 2 

 where, on account of the smallness of 0, the series converges 

 very rapidly. By writing in the above, for V and V their 

 values just given, we obtain 



dv e d*v 0* 



#- = 3=-7 + 3=TT-^ 

 dw I dw' 1.2 



In the same way, if w be a normal to B, directed towards -4, 

 and 6 t be the thickness of the glass measured along this normal, 

 we shall have 



ft -f,. 



dw 1 dw 2 1-2 



But, if we neglect quantities of the order 0, compared with 

 those retained, the following equation will evidently hold good, 



dw n dw n 



n being any whole positive number, the factor ( l) w being in- 



troduced because w and w are measured in opposite directions. 

 Now by article 4 



dV . = dV 



= ^ and 47T/> = = 



dw dw 



