XXII. On some points in the Theory of Differential Equations. By Augustus 

 De Morgan, of Trinity College, Secretary of the Royal Astronomical Society, 

 and Professor of Mathematics in University College, London. 



[Read March 27, 1854.] 



The miscellaneous character of this paper makes it desirable to prefix a brief table of 

 contents. 



1, Summary of peculiar notation and language employed. 2, Remark on self-compensat- 

 ing variables. 3, On some neglected theorems, and in particular, if either or both P 

 and Q be infinite, P:Q,~ P X :Q X . 4, On the factor of a differential, its difficulties, 

 and mode of avoiding them. 5, Reassertion of the entrance of arbitrary functions into 

 the primitives intermediate between the final equation and the general primitive, and 

 remark on the reason given by Lagrange for their rejection. 6, Connexion between 

 the ordinary and singular (whether intraneous or extraneous) solution of a primordinal 

 equation ; results of my last paper obtained without geometry ; new equation of con- 

 nexion between the usual criteria. 7, Remarkable theorem of M. Cauchy for ascertaining 

 whether a given solution be intraneous or extraneous. 8, Remark on the verification of 

 the singular solution of an equation. 9, Connexion of the ordinary and singular solu- 

 tion of biordinal equations, including the extension of Cauchy's theorem. 10, Sketch of 

 the theory of triordinal equations ; entrance of what are called determinants ; objection 

 to this name. 11, System of polar correlation. 12, Connexion of the method of 

 transformation for all differential equations given in a former paper, and here presented 

 as the method of polar transformation. 13, Polar correlation of biordinal equations. 

 14, General account of the equation <p (,r, y, %, y , «') = 0. 15, Polar correlation of 

 curves in space; limitation arising from the differential correlation being primordinal. 

 16, Remarks on Lagrange's primary solution of partial equations. 17, Primary solu- 

 tion of biordinal partial equations ; failure of result in the attempt to make self-com- 

 pensating variables of the constants. 18, Analogy of point and curve, chord through two 

 points and developable surface through two curves ; construction of a biordinal partial 

 equation. 19, Remark on the general solution of a biordinal equation. 20, Polar cor- 

 relation of surfaces ; polar biordinal equation a case of Q + Rr + Ss + Tt + U(s 2 - rt) = 0; 

 analogy of this equation with the most general biordinal equation of two variables, in its 

 primordinals, and in its singular solution. 21, Method of polar transformation for partial 

 differential equations. 22, Suggested notation for eliminants, and deduction of some pri- 

 mary properties. 23, Test of the equation Q + Rr + &c. = being polar, already arrived 

 at, from another point of view, by Ampere ; reduction of the integration to that of two 

 primordinal equations, whose common solutions are those required. 24, Remarks on 

 notation. Appendix, Convergence of Maclaurin's series, and Lagrange's theorem. 



