274 



have two parameters. The curve of equilibration is formed by a 

 perfectly flexible chain suspended in a vertical plane between two 

 fixed points. Suppose the chain loaded with heavy rods of equal 

 thickness, which hang vertically from it, indefinitely near to each 

 other, at equal horizontal intervals, and with their lower ends in a 

 horizontal straight line. It is shewn that the equation to the curve 

 is derivable from the functional equation already mentioned, and 

 that its solution in that case gives (putting y to denote the ordi- 

 natesy(.x') ) 



Amongst other curious results, the author establishes a relation 

 between the arcs of these curves similar to the properties of the 

 sines and cosines of the circle. It also appears that curves of equi- 

 libration may be constructed from the catenary, just as the ellipse 

 is formed from the circle. A series of Tables accompany the paper, 

 intended to facilitate the calculations of engineers for suspension 

 bridges or equilibrated arches. 



2. On General Differentiation. Part I. By Professor Kel- 

 land. 



" As early as the time of Leibnitz, it was suggested that a general 

 form might be discovered for dift'erential coeflBcients, somewhat in 

 the same way as that which applies to expansions. Notwithstand- 

 ing this suggestion, the difficulty of the subject appears to have 

 deterred mathematicians from engaging in it, until a few years 

 since M. Liouville published a series of memoirs, in which its prin- 

 ciples are developed, and theorems deduced applicable to the solu- 

 tion of a variety of difficult and important problems. Amongst a 

 mass of particular results, the generalization of each of which would 

 probably give a distinct aspect to the science, M. Liouville appears 

 to have selected that one which is the most readily applicable to 

 practice, and which, at the same time, affords conclusions remark- 

 ably convincing of its completeness. As far, then, as relates to the 

 foundation of the theory, nothing is wanting : when we examine 

 the formulae themselves, we find much room for improvement. In 

 the first place, not only are there different formulae for every com- 

 bination of differentials of simple quantities, but even the very 

 modes of demonstration made use of to produce them, vary widely 

 from each other. Certain of the conclusions themselves also ap- 



