14 Mr. W. H. L. Russell on Linear Differential Equations. [June 16, 



Subjoined is a Table giving the dates of the various observations, with 

 the reference numbers corresponding to those on the diagram, and with 

 remarks on the state of the sky. 



Number 

 in 



diagram. 



Date of 

 observation. 



Eemarks. 



I. 

 II. 

 III. 

 IV. 

 V. 

 VI. 

 VII. 

 VIII. 

 IX. 

 X. 

 XI. 

 XII. 

 XIII. 

 XIV. 



XV. 

 XVI. 

 XVII. 

 XVIII. 

 XIX. 

 XX. 

 XXI. 

 XXII. 

 XXIII. 



April 4th 



J anuary 8th , . . 



April 8 th 



January 9th . . . 



March 8th 



April 9th 



January 10th 

 February 9th... 

 January nth... 

 February 10th 

 January 12th... 

 November 19th 

 March 13th ... 



April 1 3 th 



April 14th 



April 15 th 



January 16 th... 

 September 20th 

 February 16th 

 April 16th 



April 1 7th 



November 22nd 

 November 23 rd 



No mention of cloud. 



No mention of cloud. 

 Extremely clear sky. 



No mention of cloud. [night by a halo. 



Sky not good ; thin hazy clouds, followed later in the 



Much wind. 



No mention of clouds. 



Occasional small clouds, and rather hazy. 



Clouds producing prismatic colours round the moon. 



Sky not good ; fleecy clouds. [clouds. 

 Bad night ; stopped after 10 minutes, in consequence of 

 Sky very clear. 



Occasional clouds. 



Sky hazy at sunset ; occasional clouds. [night. 

 Sky apparently not quite so clear as on the preceding 



Fog and white frost, afterwards drift. 

 No remark about cloud. 



XVI. "On Linear Differential Equations. 5 '— No. III. By W. H. 

 L. Russell, F.R.S. Received June 11, 1870. 



The integrals obtained in my last paper on this subject were deduced by 

 the same process which afforded the determinants in the first paper. It 

 is obvious that these integrals could be found by a more direct investiga- 

 tion. This is what I am now going to attempt. It will be found moreover 

 that the present method will have the advantage of clearing away the am- 

 biguities arising from the existence of common factors in the algebraical 

 coefficient of the highest differential, and the denominator of the exponen- 

 tial in the solution. It will also be found to lead us to certain ulterior 

 results. 



Let us take the differential equation 



+ (a'"+/3"^+y'V 2 Jy=0. 

 Let us now put in this equation 



We shall easily see that it is impossible for the exponential to contain 



