377 
connexion with the Calculus of Variations, and partly with the higher 
branches of mathematical physics, cannot well be overrated. It is well 
known that in the more advanced departments of mathematical physics 
one of the principal obstacles to our progress arises from the difficulty of 
integrating the partial differential equations which represent the condi- 
tions of the problem investigated: that these integrals have been sought in 
form of general primitives, which are in nicst cases unattainable; and 
that, if such forms of solution are attainable, a fresh difficulty arises upon 
our seeking to determine the forms of the arbitrary functions introduced. 
It is impossible now to determine what may be the physical value of the 
integrals of such differential equations, when stated in the form of com- 
plete primitives ; but, in the default of better, such solutions may afford 
us, at least, some information. The following remarks are offered in the 
hope of being thus beneficial :-— 
1, If we are given any equation, including two arbiirery constants 
a, B, and two independent variables x, y, 
S(@ Y, % 4, B)=9, (I) 
we may differentiate this equation with respect to the independent va- 
riables x and y, thus obtaining 
df 
ol 
2) 
_ and, by eliminating a, 8 between the three equations stated, we get a 
partial differential equation, in general non-linear, 
dz dz 
F (2,9, 2 re? Z)- E@ % 4 Pg) = 0, 
of which (1) is said to be a complete primitive. 
2. Again if, w being any function of known form in 2, y, 2, we were 
given an equation of the type 
Si {%, ¥, % @(u)} = 9, (1’) 
@ being any arbitrary function, we might differentiate as before, and 
eliminate ¢, 9’, thus again arriving at a partial differential equation 
dz dz 
F, (2, Y; %, dx’ 5 )=¥, (a, Y, %, P, 9) =0 
of which (I’) is said to be the general primitive. 
3. Similarly, if we are given any equation including three arbitrary 
constants a, 8, y, and three independent variables 2, y, , 
S Y, %, W, a, B, 1) = 0, (IT) 
we may differentiate this equation with respect to the independent vari- 
ables x, y, 2, thus obtaining 
B. I. A. PROC.—YOL. VII. 3H 
