370 PROCEEDINGS OF THE AMERICAN ACADEMY 



that which Euler had assigned, namely, the squares of the squares or 

 fourth powers. 



Mr. Ilodgkinson endeavored, by a careful and minute analysis, to 

 obtain fractional exponents that should express, in a single formula, 

 the relations of strength, for both difference of height and diameter, with 

 more exactness than any hitherto proposed ; and he arrived, after ex- 

 amining many other powers, at the formula ^, d being the diameter, 

 I the length, as giving the relation of strength for cast-iron pillars of 

 different sizes ; and this may be taken with perfect confidence as 

 giving the utmost strength within the limits to which experiment has 

 been carried. The above formula merely determines the comparative 

 strength of pillars of different sizes and proportions. To obtain the 

 actual sti-ength of any pillar, Mr. Hodgkinson, finding the diameter d, 

 in inches, and the length, I, in feet, and taking, for pillars with round 



ends, 14.9 for a coefficient, gives 14.9 -^ =. W, the weight in tons 

 that the pillar will just break under; and changing the coefficient 

 14.9 to 44.7, he obtains the weight that will break pillars with square 

 or flat ends. These coefficients were obtained by him as a mean, from 

 a careful comparison of all his experiments, and appear to be, as I 

 have before said of his exponents, sufficiently near the truth to be 

 relied upon for all iron of good quality. At the same time, I think 

 we ought always to be aware of the caution given by Biot, and " trust 

 no such formula much beyond the light of experiment." "While Hodg- 

 kinson thus gives a much more exact formula than that of Euler for 

 cast-iron pillars, he retains that of Euler for wooden columns, as pref- 

 erable to his own. 



Besides this course of experiments upon solid pillars, he made a 

 very good series upon those formed hollow. These ranged through 

 diameters from 1| inches up to 3| inches, the thickness varying from 

 .28 to .33 inch ; the same length, namely, 7-J- feet, being taken in all 

 cases. The greatest breaking weight used was 50,477 pounds. These 

 experiments seem to have been as well contrived and executed as 

 those upon solid pillars ; and while they show with more exactness than 

 had before been given the vast increase of strength obtained from a 

 given quantity of iron by casting it in the hollow, rather than in the 

 solid form, they did not reach up to sizes large enough to determine 

 beyond a doubt, from this law of increase, a rule which may be per- 



