24 REPORT—1883. 
ing upon Galileo we have Isaac Newton (1643-1727) : the ‘ Philosophize 
naturalis Principia Mathematica’ known as the ‘ Principia’ was first pub- 
lished in 1687. 
The physical, statical, or dynamical questions which presented them- 
selves before the publication of the ‘Principia’ were of no particular 
mathematical difficulty ; but it is quite otherwise with the crowd of 
interesting questions arising out of the theory of gravitation, which, 
in becoming the subject of mathematical investigation, have contributed 
very much to the advance of mathematics. We have the problem of two 
bodies, or, what is the same thing, that of the motion of a particle about a 
fixed centre of force, for any law of force ; we have also the (mathematically 
very interesting) problem of the motion of a body attracted to two or 
more fixed centres of force; then, next preceding that of the actual solar 
system—the problem of three bodies ; this has ever been and is far beyond 
the power of mathematics, and it is in the lunar and planetary theories 
replaced by what is mathematically a different problem, that of the 
motion of a body under the action of a principal central force and a 
disturbing force: or (in one mode of treatment) by the problem of 
disturbed elliptic motion. I would remark that we have here an instance 
in which an astronomical fact, the observed slow variation of the orbit 
of a planet, has directly suggested a mathematical method, applied to 
other dynamical problems, and which is the basis of very extensive 
modern investigations in regard to systems of differential equations. 
Again, immediately arising out of the theory of gravitation, we have 
the problem of finding the attraction of a solid body of any given 
form upon a particle, solved by Newton in the case of a homogeneous 
sphere, but which is far more difficult in the next succeeding cases of the 
spheroid of revolution (very ably treated by Maclaurin) and of the ellipsoid 
of three unequal axes: there is perhaps no problem of mathematics which 
has been treated by as great a variety of methods, or has given rise to so 
much interesting investigation as this last problem of the attraction of an 
ellipsoid upon an interior or exterior point. Jt was adynamical problem, 
that of vibrating strings, by which Lagrange was led to the theory of the 
representation of a function as the sum of a series of multiple sines and 
cosines; and connected with this we have the expansions in terms of 
Legendre’s functions P,,, suggested to him by the question just referred to 
of the attraction of an ellipsoid; the subsequent investigations of Laplace 
on the attractions of bodies differing slightly from the sphere led to the 
functions of two variables called Laplace’s functions. I have been speak- 
ing of ellipsoids, but the general theory is that of attractions, which has 
become a very wide branch of modern mathematics ; associated with it 
we have in particular the names of Gauss, Lejeune-Dirichlet, and Green ; 
and I must not omit to mention that the theory is now one relating to 
n-dimensional space. Another great problem of celestial mechanics, 
that of the motion of the earth about*its centre of gravity, in the most 
