VOL. LI.] PHILOSOPHICAL TRANSACTIONS. SIQ 



1 nearly when the effect is a maxiirium. And so, in like manner, when 

 b, m and n are given, and a is made variable, it is easy to see that, instead 

 of the load, the best distance of the power from the fulcrum of the lever 

 will be the result of the process ; viz. a -.h \\n -^^ V n'-\-mn : m. But, this by 

 the way. 



In the proportion here determined, the power w is a weight, and therefore 

 >fna — nh^ which is the generating force, being partly employed to overcome the 

 inertia of the quantity of matter /«, it is not wholly taken up in giving motion to 

 the weight n ; and the relative velocity is continually decreasing. But on the 

 other hand, if m be the force of a spring, as is that of our atmosphere, or if n 

 can be uniformly accelerated any how, in repeated vibrations, that there may be 

 no sensible diminution of the relative velocity, the whole will be exerted on the 

 weight to be raised ; i. e. the tension of the rope or chain, by which the power 

 is confined to act on the weight, will always be the same as though the beam 

 were at rest ; and then, by expunging nic^ out of the expression for the greatest 



effect, ?zv/—^ — n becomes evidently enlarged ton\/ — 7^—. The consequences 



are these. 1st, The greatest effect of this engine when m is a spring, will al- 

 ways exceed the contemporary effect where m is a weight. 2dly, The proportion 

 of the power and weight will then he n : m : : a : 2l, as appears by taking the 



fluxion of n V — t^ — = O, and reducing the equation in the manner above. 

 Whence the load to be raised for the greatest effect of a steam-engine, if the 

 inertia of the materials composing its working parts be put out of the question, 

 will be just half of what is sufficient to balance the atmosphere, whether the 

 brachia of the lever be equal or not. 



Mr. B. here adds 2 or 3 remarks on what he formerly laid down concerning 

 the proportion of the cylinders. And, 1 st, in all values of the brachia, with 



regard to their lengths, and all values of 7?, the expression >/ ^ for the time 



of a stroke, when tw is a weight, is the general expression to be used for the 

 time. 2dly, m being considered as a spring, the time of a stroke is as 

 a/ — '- — - ; and then if, according to what he there directed, a be taken variable, 



ma — no ^ 



and m the reciprocal of a, the advantages to be gained by the breadth of the 

 cylinder can only arise from a diminution of friction, and from the matter in the 



beam ; for, the expression V — — — becomes constant, and thence the strokes 



are isochronal. I might, furthermore, says Mr. B. proceed to examine into 

 these advantages, more explicitly than is there done, on the principles laid 

 dav^n, when w is a weight. But many particulars (such as the form of the bra- 

 chia and various appendages, with their quantities of matter and centres of gy- 

 ration) being wanting to perfect the theory of the construction, I shall drop the 



