TRANSACTIONS OF SECTION A, 745 
will satisfy 
pyry= (=| 2D estas moty. 2s = 1} ange -. (B) 
or (r>m) 
ope [2s 2 eas) Fe a 21 faye) 
in which D=2 fy and the usual factorial notation, viz. : 
te 
[O\*=6 (@—1) (0-2)... (@-a+1) 
is adopted. 
By the process employed in the above cited paper we are also led to the follow- 
ing results :— 
Any root of 
ary™ + by"+c=0.... (©) 
will satisfy 
D Yr m—r n eee nm — gees) m1] ter Me ee 
pr [*"p+"] y (—) Ser r TF ay 
or (r>m) ners 
T qrom oul 
Dl'y"=(— ae & aM ‘lt [""p- 2-1] ar sh te C’ 
Pit = (=) pearl -Dacat ; = ayn « » (CD 
And any root of 
ay +bay"+cxr=0.., (a) 
will satisfy 
Pees] Roufse-ta ero 
or (7 >m) 
[™p- faa sk Ber -) PD 1 eyn -.. (D) 
) em 
The complete integral of each of the above differential equations is of the form 
CY s" + CQYo™ 2 os FCmYm'y 
or CY" $CoYo” 2 ow +O Ys 
according as m or 7 is the greater, and ;, Yo,» + © Yin OF Yr are the m or r roots of 
the connected algebraic equation. 
The same results may be obtained by suitable substitutions, or interchanges, 
and reduction by known theorems. Thus (a), (A), (A’) may be changed into (0), 
(B), (B’) respectively by the substitution 
(“ 6,0, m7 ) 
c, a, b, —r, m—r 
or into (¢), (C), (C’) respectively by the substitution 
2, Overman, “4 ) 
b,c, a, r—m, —m 
or into (d), (D), (D’) respectively by the substitution 
(% Cow a7ity Ns En :) 
Cc, a, —M, T—M, L- 
Or (4), (B), (B’) may be changed into (ec), (C), (C’) respectively by means of the 
interchanges 
(22). 
3A 2 
