152 
ME. A. C. DIXON ON SIMULTANEOUS 
will be to find conditions that must be satisfied by the equations of the solution and 
to put these conditions in a convenient form for solution by inspection. 
I should add that I am greatly indebted to the referees for their suggestions and 
for help in removing obscurities. 
To the list of authorities given by Dr. Forsyth (‘ Theory of Differential Equations,’ 
Part I., pp. 299, 331), may be added the following :— 
Julius Konig. Math. Annalen, vol. 23, pp. 520, 521. 
Leo Konigsberger. Crelle, vol. 109, pp. 261-340A Math. Annalen, vol. 41, 
pp. 260-285.f Math. Annalen, vol. 44, pp. 17-40. 
Ed. y. Weber. Munchen Ber., vol. 25, 423-442. 
J. M‘Cowan. Edinb. Math. Soc. Proc., vol. 10, 63-70. 
Hamburger. Crelle, vol. 110, pp. 158-176. 
C. Bourlet. Annales de lEcole Normale (3), vol. 8. 
Biquier. Comptes Bendus, vols. 114, 116, 119. Annales de lEcole Normale 
(3), vol. 10. 
Lloyd Tanner. Proc. Lond. Math. Soc., vols. 7-11. 
J. Brill. Quarterly Journal of Math., vol. 30, pp. 221-242. 
Several of the above papers are only known to me through abstracts. 
On Bidifferentials. 
§ 2. The idea of a “ complete differential ” plays an important part in the theory 
of differential equations. In this paper I shall try to show the importance of an 
extension of the same idea to differential elements of higher orders, such as enter 
into multiple integrals. 
An expression Xdx + Y dy is called a complete differential when X, Y are functions 
of the independent variables x, y, such that 
dY/dx = dXj?ny. 
If this is the case, then, under certain restrictions, the value of j(X<Ar -f- Y dy) depends 
only on the limiting values of the variables, and not on the intermediate ones by 
which these limits are connected, or, as generally expressed, on the path along which 
the integral is taken. 
Co 
This depends on the theorem that 
jew* + Yd,,) = fj'dr-lr)^ 
* For reasons stated below, I am not in agreement with the results given in the latter part of this 
paper. 
f In this paper it should be noticed that the equations (52) on p. 266 are not more general than (46). 
