6 
the others are derivable from it. It is to be noticed that what 
is thus found out is an unknown quantity. In an analogous 
manner, the solution of a differential equation containing two 
variables, determines an unknown relation between the variables 
in the form of an algebraic equation, involving the variables 
together with arbitrary constant quantities introduced by the 
rules of the solution. This equation expresses the relation that 
subsists between the variables under every change of their 
actual values, and is, in fact, the answer which it was proposed 
to obtain by forming the differential equation. It was virtually 
by this process that Newton proved that the form of the orbit 
of a planet is given by the equation of a conic section. By 
having the arbitrary constants at disposal, the abstract solution 
may be made to apply to an actual instance. For example, a 
few observations such as those which Kepler employed to 
determine the form of the orbit of Mars, would suffice to fix 
very approximately the arbitrary constants in the analytical 
solution, and thereby obtain that equation of the planet's elliptic 
path which Kepler deduced with so much labour from a very 
large number of observations. In physical astronomy we have 
often to deal with equations involving more than two variables ; 
but in such cases the number of the variables is always one 
more than the number of the equations, so that the several 
equations are reducible to a single one involving only two of 
the variables. 
8. But in physical science problems come before us of such 
kind that the single differential equation to which the several 
differential equations formed to express the given conditions 
of a proposed question are reducible^ contains not fewer than 
three variables. The problems I refer to relate to phenomena 
of light, heat, electricity, and magnetism. The analytical solu- 
tions of equations that contain three or more variables, and 
the applications of the solutions in answering questions of the 
above-mentioned classes, constitute an advance in physical 
theory of the same kind as that which was made when the 
solutions of equations containing two variables were applied in 
physical astronomy. But on account of the greater compre- 
hensiveness of the equations, and complexity of the conditions 
which their solutions have to satisfy in order to account for 
experimental facts, the ansAvers to these questions are attended 
with difficulties, which, hitherto, can only be said to have been 
partially overcome. It is certain, however, that if physical 
science bo something more than the certifying of facts and laws 
by experiment, and if, in order to bo complete, it must be 
capable of accounting for experimental facts and laws by 
reasoning based on definite and intelligible principles, there 
