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IV. On Simultaneous Partial Differential Equations. 
By A. C. Dixon, Sc.D. 
Communicated by J. W. L. Glaisher, Sc.D. 
Received May 9—Read June 15, 1899. 
§ 1 . 
5 § 
-9. 
§§ 10-13. 
§§ 14—30. 
31—42 
Contents. 
Pages 
Introductory .151 
On “bidifferentials,” or the elements of double integrals, and on the conditions to be 
satisfied in order that a given bidifferential expression may be a complete bidiffer¬ 
ential . 152—159 
Theory of equations linear in the Jacobians of two unknown functions; their solution 
reduced to the formation of complete bidifferentials.159—162 
Theory of other simultaneous partial differential equations in two independent and 
two dependent variables. A method of solution, with examples of its application. 
One pair of variables is said to be a “bifunction” of other pairs when its bidifferen¬ 
tial can be linearly expressed in terms of theirs: this idea is of importance in con¬ 
nection with the derivation of all possible solutions when complete primitives are 
known. Construction of bifunctions in some cases.162—181 
Differential equations of the second order with one dependent and two independent 
variables. A method of solution, with examples.181—191 
§ 1. In this paper, without touching on the question of the existence of integrals 
of systems of simultaneous partial differential equations, I have given a method by 
which the problem of finding their complete primitives may be attacked. 
The cases discussed are two : that of a pair of equations of the first order in two 
dependent and two independent variables, and that of a single equation of the second 
order, with one dependent and two independent variables. 
I follow, as far as possible, the analogy of the method of Lagrange and Charpit, 
and with this object introduce the conception of the “ bidifferential ” or differential 
element of the second order, which bears the same relation to a Jacobian taken with 
respect to two independent variables as a differential does to a differential coefficient. 
The solutions considered are, in general, complete primitives, that is, such as contain 
arbitrary constants in such number that the result of their elimination is the system 
of equations proposed for solution. The existence of such primitives is sufficiently 
established (see the papers of Frau von Kowalevsky and Professor Konigsberger, 
quoted hereafter) ; it will therefore be assumed, and the object of the investigation 
5.11.1900 
