Prof. Doiikiu on a Class of Differential Equations. 47 



" On a Class of Differential Equations, including those which 

 occur in Dynamical Problems." — Part. II. By W. F. Donkin, M.A., 

 F.R.S., F.R.A.S. &c. 



This is the second and concluding part of a paper of which the 

 first part was printed in the Philosophical Transactions for 1854. 

 In the fourth section (the first of this part) some of the most import- 

 ant results of the former part are recapitulated. 



In the fifth section the theory of the Variation of Elements is -con- 

 sidered under that aspect which belongs to it in connexion with the 

 general methods of this paper ; and the facility of its application is 

 shown in two instances : ( 1 ) the expressions for the variations of 

 the elliptic elements of a disturbed planet's orbit are deduced from 

 the results of Art. 30 (Part I.), on undisturbed elliptic motion ; 

 (2) the problem of determining the motion of a free simple pendulum 

 (omitting the effect of the earth's rotation) is treated by considering 

 the orbit of the projection of the bob upon a horizontal plane as a dis- 

 turbed ellipse. The differential equations which define the variations 

 of the elements of the ellipse are given in a rigorous form, and inte- 

 grated approximately so as to give the motion of the apsides of the 

 mean ellipse in any case where the pendulum never deviates much 

 from the vertical, and the motion is not very nearly circular. The 

 result agrees with the conclusions of the Astronomer Royal (Pro- 

 ceedings of the Royal Astronomical Society, vol. xi. p. 160). 



In the fifth section the transformation of the differential equations 

 by the substitution of new variables is considered, and particularly 

 that, kind of transformation, called by the author a normal trans- 

 formation, which leads to a new System of equations, not merely 

 possessing the same general form as the old, but distinguished also 

 by other common properties. A definition is given of tliose trans- 

 formations which may be })roperly called, from analogy, transforma- 

 tions of coordinates, and it is shown that all transformations of coor- 

 dinates are normal. General formulee are given for transforming the 

 equations of any dynamical problem horn, fixed or moving systems of 

 axes of coordinates ; and an illustration is drawn from the case of the 

 motion of a planet referred to axes in the varying plane of its own 

 orbit. 



In the seventh and last section the principles of transformation 

 developed in the preceding section are applied in a more general 

 manner to the differential equations of the planetary theory ; and it 

 is shown that when the motions of a planetary system are referred 

 to a system of rectangular axes having their origin in the sun, and 

 otherwise moving in any arbitrary manner, tlie variations of the ele- 

 ments will still be determined by the same formulse as if the axes 

 were fixed, provided there be added to the disturbing function R, for 

 ' each planet, the expression 



'v//xa(j — e'').(w„sin V sini — Wj cos v sin<-J-c<;^, cos j), 



in which i is tlie inclination of the orbit to the (moving) plane of xy, 

 the longitude of the node reckoned from the axis of x, and Wy, w,, w.^ ; 

 the angular velocities of the moving system of axes about the three 



