226 Royal Society. 



of times, or may be subdivided into any number of equal parts, by 

 finite algebraic equations. What he had accomplished with respect 

 to the arcs of the lemniscates, which are expressed by a particular 

 elliptic integral, Euler extended to all transcendents of the same class. 

 Landen showed that the arcs of the hyperbola may be reduced, by a 

 proper transformation, to those of an ellipse. Lagrange furnished us 

 with a general method for changing an elliptic function into another 

 having a different modulus ; a process which greatly facilitates the 

 numerical calculation of this class of integrals. Legendre distributed 

 the elliptic functions into distinct classes, and reduced them to a re- 

 gular theory, developing many of their properties which were before 

 unknown, and introducing many important additions and improve- 

 ments in the theory. Mr. Abel of Christiania happily conceived the 

 idea of expressing the amplitude of an elliptic function in terms of 

 the function itself which led to the discovery of many new and useful 

 properties. Mr. Jacobi proved, by a different method, that an elliptic 

 function maybe transformed in innumerable ways into another similar 

 function, to which it bears constantly the same proportion. But his 

 demonstrations require long and complicated calculations ; and the 

 train of deductions he pursues does not lead naturally to the truths 

 which are proved, nor does it present in a connected view all the 

 conclusions which the theory embraces. The author of the present 

 paper gives a comprehensive view of the theory in its full extent, and 

 deduces all the connected truths from the same principle. He finds 

 that the sines or cosines of the amplitudes, used in the transformations, 

 are analogous to the sines or cosines of two circular arcs, one of which 

 is a multiple of the other; so that the former quantities are changed 

 into the latter when the modulus is supposed to vanish in the alge- 

 braic expression. Hence he is enabled to transfer to the elliptic 

 transcendents the same methods of investigation that succeed in the 

 circle : a procedure which renders the demonstrations considerably 

 shorter, and which removes most of the difficulties, in consequence of 

 the close analogy that subsists between the two cases. 



A paper was read, entitled, "An Experimental Investigation of the 

 Phaenomena of Endosmose and Exosmose." By William Kitchie, Esq., 

 M.A., F.R.S., Professor of Natural Philosophy in the Royal Institu- 

 tion of Great Britain, 



Mr.Porret had, in the year J 81 6, announced the discovery, that if a 

 vessel containing water be divided into two compartments by a dia- 

 phragm of bladder, and placed in the voltaic circuit, the water would 

 rise on the negative side above its level in the positive compartment. 

 M. Dutrochet discovered, that if .ilcohol be placed in one of the cham- 

 bers, and water in the other, without employing the voltaic battery, 

 the water will percolate through the bladder, and the fluid will rise in 

 the chamber containing the alcohol : an action to which he gave the 

 names of Endosmose and Exosmose, according to its direction with 

 regard to the side of the membrane considered ; comparing its two 

 sides to those of a Leyden jar in opposite electrical states. This 

 electrical theory has been combated by M. Poisson : but the true expla- 

 nation of this singular phaenomenon does not a[)pear to have been yet 

 ariven. 



