1869.] Mr. W. H. L. Russell on Linear Differential Equations. 119 



When a, j3, y are none of them equal to zero, and 



p 2 y— py'+y"=Q> 



P r =p 2 /3-{2 7 (r-l) + /3'} P + (r-l) 7 ' + /3', 

 Q r = p 2 a - (2/3r + a')p -f 7 r(r - 1 ) + fi'r + a", 



Rr=(r+l)(--2|t>a+ j Sr+a') J 

 S r =a(r+l)(r+2). 



There will be (ft + 2) horizontal and (ft -4-1) vertical rows, where n is the 

 index of the highest power of x in the denominator of <p. 

 When a and (5 are not zero but y=0, and we put 



then 



F r = £fx 2 + 2a/* v - /3> - (r/3 + a> + /3", 

 Q r =a M ? -(2/3r + a>-(2r+l)«^ + a" + r/3', 

 R r =(r+l)( / 3r-2a ( u + a'), 

 S r =a(r+l)(r+2). 



When a is not zero, but j3 = y=0, and we put 



a' y" j3' y' 



ay a a 



P r =ccjiv — aV— a(r+ l)o + /3", 

 Q r = a/x 2 — a 7* — >u (r -f 1 ) + a", 

 R r =(r+l)(«'-2a#, 

 S r =a(r+l)(r4-2). 



Similar methods will apply to the linear differential equation 



O + ^+y^ 2 )^ +(a ' + / 3^ + yV) ^ +( a " + /3".T + yV)^ 



+(a'"+i3" , ar+y"V)a=0, 



and the process admits of a very remarkable simplification. All linear 

 differential equations of the second and third orders may be treated in the 

 same way, and, I believe, all linear differential equations of every degree*. 



V. "Spectroscopic Observations of the Solar Prominences, being Ex- 

 tracts from a Letter addressed to Sir J. F. W. Herschel, Bart., 

 E.R.S., by Captain Herschel, R.E., dated ' Bangalore, June 

 12th and 15th, 1869/ " Communicated by Sir J. Herschel. 

 Received July 19, 1869. (See p. 62.) 



* This investigation assumes that a+fix+yx 2 and the denominator of <p hare no 

 common factor.— W. H. L. R., Jan. 13, 18G9. 



