28 Proceedings of the Royal Irish Academij. 



an additional load along the rib, a tifteenth or lesser fraction of r, it is better 

 to get rid of the i?- load along the rib altogether, and restore siich fraction 

 of it as may be necessary later on. 



To find the conjugate load for a vertical load uniform along a circular 

 quadrantal rib, v.e proceed by the calculus, as Eankiue does for the spandril 

 area. In doing so for load \r it has only to be done once and for all, and 

 can be adjusted by simple proportion for lesser thicknesses. "We have always 

 6 = P cot Q, and Q is split up into thin horizontal strips q^.dy, where y is 

 measured from H up to L : and P is divided into thin vertical strips i}.dx, 

 where x is measured from A towards E. "We then have fig. ha — 



5 = "^ = I- (P cot i) ; P = ^ . 5 = ^ . ri = J i ; P cot i = '- i cot i; 

 ^ ay ay I A A A 



d ?•■ ■ '^y " i 



^ . {P cot i) = K- (cot i - i cosec^ i); y = r cosi: — = - r sin i ; 



and ^ = ^ — :. So that the value oifv is Q = -r (P cot i) 

 dy r sm i •' "^ dy ^ 



rf , „ , -^ di 1^ , , ■ ■ - -^ 1 sin i cos i - i 



= -p (P cot i) T- = 77 (cot I - I cosec- t) — -. — : = r =r—~r-. — , 



(h^ ' dy 2 ^ ' r sin % 2 sm' % ' 



a part of Eankine's expression. He failed to notice that the other part was 

 the right-angled isosceles triangle we have described, or he would not have 

 proposed to build the conjugate area in the unsatisfactory way he does. 



From the modified form q = fv = ^p-^ — ^^ — -. — — r the ten breadths of 



^ • o sm I - sm oi 



abkd, fig. DO., are calculated and plotted ; the top value is found to be ^r 

 when i = zero, iu the usual way, by differentiation of the numerator and 

 denominator of q separately. The upward thrust at the springing £ is the 

 area of the collar or the quadrant, and equals ^ tt ?" ; and this divided by 

 the radius gives the value of the normal loadat P as kd = ^ tt r = •7854?\ Again, 

 the normal load at the crown is |/-, the depth of the collar. Multiplying 

 this by r, we have the horizon area abl-d = ^r, just as for a load ir spread 

 along the span ; but now, instead of being also ir spread along ad the rise, it 

 begins by ab = -^r, and ends with dk, a trifle greater than f »•, and as the 

 average is ir the boundary from b to k is slightly curved. The curve we 

 replace by the trchlc-battcrbuiindary, where Ig batters at 1 iui for four-ninths 



two scientific men interested in it, it appears as if the theorem were new to hydrost-atics. 

 From criticism of the method of conjugate load-area.s employed, it seems as if this 

 elegant method, especially lending itiself to graphical construction, were little knonn or 

 understood. The theorem can be proved by the strict but laborious methods of the 

 integral calculus. 



