60 



the terras depending on the cubes of the eccentricities and quantities 

 of that order, which he does by means of a table, similar to the one 

 given in his lunar theory; and applies them particularly to the deter- 

 mination of the great inequality of Jupiter, or at least such part of it 

 as depends on the first power of the disturbing force. That part which 

 depends on the square of the disturbing force may, he thinks, be most 

 easily calculated by the methods given in his lunar theory. He re- 

 commends it as particularly convenient to designate the arguments 

 of the planetary disturbances by indices. The bulk of the paper is 

 occupied by the tables, and by examples demonstrating their use. 



A paper was read, " On the Theory of the Elliptic Transcendents." 

 By James Ivory, A.M., F.R.S., &c.' 



Fagnani discovered that the two arcs of the periphery of a given 

 ellipse may be determined in many ways, so that their difference shall 

 be equal to an assignable straight line; and proved that any arc of 

 a lemniscate, like that of a circle, may be multiplified any number 

 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, Eulerextended toall 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 Christiana 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 ma} r be transformed in innumerable wa} 7 s 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. 



