846 Transactions of the American Institute. 



engines which I have built have been constructed upon that principle. 

 I am unspeakably gratified that the investigation by the calculus, of 

 which I am ignorant, brings investigators to the same results. 



It is undoubtedly possible to have the counterbalancing weight too 

 large. Without any counterbalancing weight, the engine would be 

 unsteady. The recoil of the cylinder, in expelling the piston, being 

 precisely the same as that of a gun expelling the shot, must be neutral- 

 ized. In our larger engine, we have found that one half the weight 

 of the reciprocating parts is sufficient to keep the engine steady ; but 

 in the smaller size we have had to make the counterbalance equal in 



weight to the whole reciprocating parts ; for otherwise they would 

 not run with perfect steadiness. 



Mr. J. K. Fisher — My impression is that we are beginning to under- 

 stand this subject more clearly. The utility of high speed engines, 

 and of counter- weights, on locomotives, is recognized ; and it is 

 understood that in cutting off short, the strain is not at the beginning 

 of the stroke, but at the end of the stroke. The men who build 

 engines are not generally highly educated mathematicians, and we 

 need to have mathematicians calculate the distribution of the power, 

 and the strain upon the materials. All our formulae for strains are 

 false until the time of Rankin. When Watt began to work with his 

 steam-engine without a cut-off, he found that his piston came down 

 with a bang, and by experiments he found that he could do the work, 

 cutting off at one-quarter of the stroke, without increasing the press- 

 ure of the steam. The use of the steam expansively has saved an 

 enormous amount of fuel, and prevented great waste of power. I 

 am very glad to see professional mathematicians co-operating with 

 practical men, for they are needed as writers. 



On motion of Mr. C. E. Emery, the thanks of the Association were 

 unanimously tendered to Dr. Barnard for his ve/y able paper pre- 

 sented this evening. 



Mr. Hamilton E. Towle — It is usual in theaters, after a heavy 

 tragedy to have a little farce ; and, if it is not inappropriate, I will 

 give the mathematicians present the benefit of a new symbol which 

 met with some approval the first time it came before an audience. 



It has been remarked that the formulas of science do not provide 

 for all that we have to contend with in practice. By way of illustra- 

 tion, I will suppose a wall pressed on two sides by different heads of 

 water H' and H', as in the case of a lock-gate. I will not trouble . 

 you with differentiation or integration ; but there are formulas which 

 determine that under such circumstances the thickness must be 



