4418 Determination of the Thickness of Wail [JUNE 
SECOND PRINCIPLE. 
The variation of an integral formula is equal to the integral of 
the variation of its differential ; that is to say, 0(/%) = f(@é). 
Let f§ = x, and consequently € = dx; hence by taking the 
variations 0 = 3(dx%) = d (dx). Integrating this last equation, 
we shall have (32) = 0% = d(/f2). 
CoroLitary.—Therefore in repeated integrations we may change 
the signs / and dat pleasure ; for we have found ¢(/2) = f(9&) ; 
therefore (ff) = f(3s2) = Sf (7 2). 
Similarly 7 SSS) =SOSSD =SS CSD =SLS OD, &e. 
PROBLEM. 
To determine the variation of any indefinite integral formula 
fildaxr:— 
Whatever the function II may be, we have always by the second 
principle 0(fI1da) = fo(Md«x); but d0(I1dax) = dx dTl + 
113d ax; and the first principle gives 0 (d x) = d (II); therefore 
sfidx =Sfdxtil + /tlddax. But by the method of inte- 
grating by parts,* the last term fMdde = Woda —fdilex; 
and therefore by substitution 7/1 da = Wx + fdexdtl — 
fd dx; ord f/Idx =Nee +f (ded —diIlex), 
ARTICLE V. 
Determination of the Thickness of Wall necessary to support a given 
Arch. By Mr. James Adams. 
(To Dr. Thomson.) 
SIR, Stonehouse, April 20, 1817. 
Ir in your opinion the following question and solutions merit a 
place in the Annals of Philosephy, your inserting them therein 
will much oblige your humble servant, 
rs JAMES ADAMS. 
a 
The Question. 
ABHD (PI. LXVIL. Fig. 9) represent a vertical section of half a 
brick arch and work over its top, and DE FG a vertical section of 
the perpendicular side wall of brick also, the slope E H being 
parallel to the chord AB; the angle B A O = 30°, semi-span 
* Since by the theory of differentials d. ry = xyd+ yd 2, integrating and 
transposing, it becomes f'xdy=2ry— fydx, which is the general formula 
for integrating by parts, 
