IV. On Certain Linear Differential Equations of Astronomical Interest. 
By Prof. H. F. Baker, F.R.S. 
Received May 13,—Read June 17, 1915. 
Contents. 
PART I. 
Page 
§ 1. Remark as to a periodic solution of a certain linear differential equation of the second order 
with periodic coefficients.130 
§ 2. Calculation of characteristic exponents for a linear differential equation of the second 
order, whose coefficients are power series in a parameter with periodic terms . . . . 132 
§ 3. Application to the Mathieu-Hill equation in general, with an example.134 
§ 4. Elementary method for computation in the case of the equation for the lunar perigee . . 137 
§ 5. Stability of the oscillation represented by the Mathieu-Hill equation, for one case . . . 141 
§6. Method for general case. 141 
§ 7. Formation and discussion of the equations for the small oscillations of three gravitating 
particles about the angular points of an equilateral triangle.144 
§ 8. Brief consideration of the solution by means of an infinite determinant.149 
§ 9. Computation of solution by means of series.151 
§10. Practical case.154 
PART II. 
§ 11. Question raised by Poincare whether the periodicity of the coefficients in a certain 
differential equation is necessary for the convergence of the solution.154 
§12. Exposition of a general method for the solution of systems of linear differential equations 
in a form valid for an indefinitely extended region.155 
§13. Application to the particular case.discussed by Poincare.160 
§14. Application to equations with periodic coefficients. Simplification of Laplace’s general 
method for avoiding explicit occurrence of the time.162 
§15. Applications of the method to the general equations of dynamics.164 
§16. Same continued.166 
§17. Same continued. 170 
§ 18. Simple example of the application of the method to computing the solution of an equation 171 
§19. Detailed application to computing the solution and characteristic exponent for critical 
cases of the Mathieu-Hill equation.173 
§ 20. Same continued.179 
.§ 21. Same continued. [Conditions for stability in general].182 
PART III. 
§ 22. Remark as to generalisation of the results of this paper.185 
VOL. CCXYI.-A 541. T [Published January 5, 1916. 
