478 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



"The use of hyperbolic logarithms is indispeusable to the computation of 

 the exact power of an engine working expansively, and as tables of such 

 logarithms are rarely in possession of practical men, we here give such an 

 assortment as will suffice for the uses of the engineer." 



Then, after describing the various kinds of boilers in use by the Cornish 

 engines, he adds page 86: 



" It will be seen, from what we have already advanced, that but a small 

 part of the superior duty of the Cornish engine can be derived from the 

 boilers; we must therefore look to the engines for the principal sources of 

 superiority, which may be comprised under three heads: 



First. The use of high pressure steam, cut off when a very small part of 

 the stroke has been performed, and working expansively over the remainder. 

 Second. The careful clothing of every part of the engine where heat can 

 escape, jacketing the cylinder, &c., &c. 



Third. The main source of the great duty is to be found in the excellent 

 system of registering and publishing the duty of each engine, which has 

 prevailed in Cornwall for many years. It has made both the proprietors 

 and engineers much more careful of a host of details that have elsewhere 

 been considered too trifling to require notice, but which, in the aggregate, 

 are of no small importance. 



"The grand secret, however, of the economy of the Cornish engines lies 



in the large application of the principle of expansion, and the results there 



obtained are very little aided by any peculiar excellence in the boilers." 



Then, al^cr describing several engines and comparing point of cutting 



off and relative economy, he adds: 



" Other causes may influence these comparisons, especially the last 

 example, where one engine is a double acting rotative engine, and the 

 other a single acting pumping one; but there is no doubt that the expan- 

 sive action in the latter is the principal cause of its more economical 

 performance. 



" The mechanical effect of expansion is most readily determined by the 

 principles of the integral calculus, and we prefer a recoui\se to that analy- 

 sis on the pi-esent occasion. Steam, like all other gases, follows the law of 

 " Manotte," which may be stated in the following words: "The volume of 

 a given quantity is inversely proportional to the pressure to which it is 

 subjected. Before this law can be rigorously applied to steam, it is neces- 

 sary to suppose the temperature constant throughout the whole expansion; 

 a supposition which, though not strictly correct, is so nearly true that we 

 may reason upon it without material error. 



" We must, however, dismissing these general considerations, now show 

 the method of calculating the mechanical effect gained by this method of 

 working steam engines, which is one of the most important topics of which 

 we have to treat." 



After giving the rule to find the increased efficiency, by working steam 

 expansively, he gives example. " First, suppose that the pressure of the 

 steam working an engine is 45 pounds on tlie square inch above the 

 atmosphere, and that the steam is cut off at one-fourth the stroke, what is 

 the increase of efficiency due to this measure of expansion ? If one-fourth 

 be reckoned as one, then four-fourths must be taken as four, and by refer- 



