Dr. Hargreave on the Problem of Three Bodies. 467 



cesses of this nature principally in this particular, that the variations 

 are represented in explicit terms of the elements themselves and of 

 the time, and not through the medium of partial differential coeffi- 

 cients. It has been his object to render the processes as elementary 

 as possible ; and to preserve them in a rigorous form, by post- 

 poning all attempts at approximation until the formulae are actually 

 applied to practical problems. The applications given in the paper 

 comprise the circular and spherical pendulums, and the planetary 

 and lunar theories, and a special theorem as to the movement of 

 the plane of a planet's motion under the influence of several other 

 planets. 



The original normal problem which is taken as the basis, is that 

 of motion about a fixed centre of force, where the force is directly as 

 the distance ; or, in other words, the system of equations not ex- 

 ceeding three in immber, of the form 



whose solutions are represented under the form 



x=\aaco^(nt+p) +fiab sin {nt+p), 

 y=\b a cos int + p) +fib b siu {nt+p), 

 z=Xc a cos (nt+p) + fig b sin (nt + p) ; 



where 



Xa= cos (j> cos \l/ — sin sin \p cos i, 



Xj= cos (jt sin »// -|- sin cos \p cos i, 

 \c= sin (p sin i ; 



/Xa= — sin (p cos »// — cos sin \p cos t, 

 ^j= — sin (j) sin \p + cos <l> cos tp cos t, 

 /LJe= cos</)sint; 



to which are afterwards added, 



j'„= sin \p sin i, 



vb=- — cos i// sin t, 



»/(.== cos t. 



These are the equations of an ellipse whose centre is at the force, 

 and situated in a plane inclined at the angle t to the plane of x y, and 

 the longitude of whose node is -^ ; and is the angular distance of 

 the major axis of the elhpse from the node ; a and b are the semi- 

 axes of the ellipse ; and p is the angular distance, from the major 

 axis, of the zero-point of the motion, measured on the circle described 

 on the major axis. A uniform motion arovmd the circle represents 

 the i)lace of the body by the corresponding point on the ellipse, 

 where it is cut by a perpendicular dropped on the major axis. 



If the force be not situated at the origin, but at the point (X, Y, Z), 

 wc have merely to substitute a;— X for x, &c. in the above equations 

 of motion and solutions. 



It is then shown that a system of the form 

 x'' + n-x=-Vx, Sec, 

 2 113 



