IV. 



General Properties of certain Partial Differential Equations similar to those of 



Hydrodynamics. 



BY THOMAS CRAIG, Ph. D. 

 U. S. Coast and Geodetic Survey, and Johns Hopkins University. 



Presented January 12, 1881. 



The following investigation was suggested by certain results which I obtained 

 in a previous paper entitled " General Properties of the Equations of Steady Motion " 

 which has been published by the United States Coast Survey. It was my intention 

 in that paper to study steady motion only, but as I went on with the work, I per- 

 ceived that many of the results were applicable to fluid motion in general, and that 

 the whole subject admitted of a generalization which would present a number of 

 interesting points. 



In Vol. LIV. of " Crelle's Journal" there is a paper by Clebsch entitled "liber 

 eine allgemeine Transformation der hydrodynamischen Gkichimgen," from which I have 

 taken the proofs of two properties of a certain determinant ; the proofs are also to be 

 found in Baltzer's treatise and in Scott's treatise on determinants. I did not see 

 Clebsch's paper until I had nearly completed mine, or I could have omitted one or 

 two tedious processes that I have given by merely referring to Clebsch's article for 

 the required demonstrations. It will perhaps be better, however, to leave this paper 

 as it is, both because it is simpler and because the two papers are so unlike in their 

 aims and methods of development, that, in order to merely refer to the article by 

 Clebsch, it would be necessary for me to change my article throughout. Clebsch has 

 in view merely a general transformation of the equations of hydrodynamics, while I 

 have in view a general investigation of the " higher orders of motion" (vide Note at 

 end) and of the properties of a certain set of Partial Differential Equations of the first 

 order. 



The present paper is divided into two parts, of which Part I. is herewith com- 

 municated to the Academy ; Part II. is not yet completed. In Part II. it is my 



