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XVI. 
THE PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL 
PHYSICS. Parr I. 
By A. W. CONWAY, M.A., F.R.U.I., Professor of Mathematical Physics, 
University College, Dublin. 
Read, Drcemprr 20, 1904; Published, Marcu 28, 1905. 
> ) 
CONTENTS. 
Page. Page. 
I. Introduction, . : : : o SY Y. Equations of ClassIII., . : loc 
II. Classification of Equations and Funda- VI. Non-homogeneous Equations, . » 17 
mental Principles, ° ° - 188 ) vir, Non-symmetrical Solutions, : . 198 
II. Equations of Class I., > 60 6 TS | Vanti, KGretie Sobiion, 5 6 o « 108 
TY. Equations of Class II., ; ; > gil 
I.— INTRODUCTION. 
Tue linear partial differential equations of the second order, with constant 
coefficients, are of the greatest importance in Mathematical Physics; and their 
study has given rise to some of the most important functions of analysis. The 
full development appears to have been retarded for several reasons. ‘The principal 
phenomena of physics to which they were applied were periodic; and hence 
attention was, for the most part, given to those solutions depending on circular 
functions. Again, there were no physical facts to lead to the consideration of 
singularities moving in space; and, lastly, there was the want of a general method 
of solution. 
A method of great generality has been given by Whittaker in which singular 
solutions are built up from plane-wave solutions; but the reverse order appears 
more natural. In this paper the singular solution is placed first. By a general 
method, singular solutions of all classes of these equations are found sufficiently 
general to include all hitherto known singular solutions. 
It is intended later to give the further consequences of these results as 
far as they are of interest in Mathematical Physics. Reference may be made 
TRANS. ROY. DUB. SOC., N.S., VOL. VIII, PART XVI. 21 
