Oct. 19, 1882] 



NATURE 



601 



for x 



e ; e being any number between — 1 and + 



This cannot hold for sin [x*]— J, and therefore we 



must consider E = o. The solution therefore is restricted 



to .F. cos 1 x*l-^ )■ At the centre of the bar. where 



x = o, this must = B. Therefore the solution is 



y = B.cosfxij — ). 



Now here we have a very remarkable circumstance. 

 It will be remembered that in the first investigation we 

 arrived at a relation between w, the weight, and b, the 

 greatest ordinate of the curve. But here we find no 

 relation whatever ; and we come to this conclusion, that 

 for the state of equilibrium fundamentally assumed, the 

 degree of bulge of the bar is immaterial. And this 

 agrees with plain reasoning : by varying the bulge of the 

 bar, we vary in equal proportions, (1) the elasticity which 

 depends on that bulge and on the general curvature, and 

 (2) the distance of the line of action of W from each 

 point p, and its consequent angular momentum ; and 

 therefore, if they are equal for one degree of bulge, they 



will be equal for every degree of bulge. The value of b\ 

 therefore is absolutely indeterminate. 



But we do obtain one most important conclusion. 



When x = + — , y must = o. And since, in the product 

 2 



Z?.cos( - \'-^)> we are not permitted to make B neces- 

 sarily = o, we must make cos( — V— )= o The sim- 

 plest form of effecting this is by making — v' jf = - , or 



JV= C. — . Substituting for C the value " ' '"', « liic'i was 



flr 4S . b 



found from the first investigation. 



rf = 4.?.w=o-2o6.^,.:. ; 

 48 b b 



and this defines the limiting value of the weight under 

 which the curved bar can rest. If the weight be dimi- 

 nished, the curved bar will expand and lift it ; if the weight 

 be increased, that increased weight will crush down the 

 curve. 



It is important to observe that the first and second in- 



vestigations apply to the same bar. And thus, in order 

 to ascertain the limiting buckling force, we need only to 

 ascertain by experiment on the same bar the amount of 

 bend produced by any convenient transversal force. 



In some cases, instead of making the first measure by 

 application of the weight w to act horizontally on the 

 middle of the bar, it may be more convenient to make a 

 measure of the vertical flexure of the bar (supported at its 

 two ends in a free horizontal position), produced by its 

 own weight. The following will be the corresponding 

 theory. 



(Third). Use the diagram of the first investigation, 

 but substitute c for b, and put Z for the whole 

 weight of the bar : and estimate the angular mo- 

 mentum round the point p. The reaction upwards of 



the force — at the pin produces - X x. The action 



2 2 



downwards of the weight of bar included between 



the pin and the point p, which is — , will produce 



a 



— - X - or — - — . The combination of these produces 



alia r 



the angular momentum Z (~~ — ) upwards. The elas- 

 v 2 2a' 



'Hz 

 j.i--- 



same value as in the first and second investigations. 

 (ax- x-). The first 



Making these equal, C. 'LJL = Z 



constant. At 



d x- 2 a ' 



integration gives C.' /v = i('L v : _ -L 3 ) 4. 

 dx a \ 4 6 / 



the middle of the bar, where x = -, l LZ mu st = a ■ the 

 2 d x 



constant therefore equals - ( — - —\ ■ and C.'^- = 



a \i6 48/ dx 



Z la x- .r 3 (? :: \ 

 a \ 4 6 24/ 



Integrating again, Cy = Z ('L x3 _ I_ 4 _ a lZ) +new 

 a \ 12 24 24 / 



constant. This is to be o when x = -; the constant 



Z \ 



is found to be + — . a 4 . — i . For the value when x = o, 

 a 384 



we have C.c = 5 " ' Z , or C= 



384 



Inserting this value of C in the expression 



and consequently j 



-,,-.z 



584 . e 



found in the second investigation, 



lV=Vl 5 <r ' ■ y - = 5 -" "-_ 



a" ■ 3S4 . c 3S4 ' c 



.Z = o-i 2 S2 Z, 



where (as before) //' is the limit of weight acting endwise 

 on the bar, which the bar can bear without buckling. 



If we wish roughly to introduce the consideration of 

 the bar's weight, ic will be sufficient to remark that at the 

 lower part of the bar the whole weight of the bar is acting 

 in conjunction with the weight W j and therefore, when 

 we have computed the force (as above) we ought to deduct 

 from that result the weight of the bar, and the residual 

 will be the force which is permissible for action on the 

 top of the bar. G. B. A. 



THE LATE DR. VAX MO.XCKHOVEN 



IN Dr. Ddsire Charles Van Monckhoven the scientific 

 world has lost an able coadjutor, and his death is to 

 be the more regretted in that he was taken from his 

 many friends when almost in the prime of life. Van 

 Monckhoven was born on September 25, 1834, and on 

 September 25 of this year he died, having thus only 

 traversed forty-eight years of the threescore-and-ten years 



