38 Mr. J. Cockle on a New Species of D\ffei°ential Equations. 



that is to say, forms not recognized as primary in Dr. Boole's 

 theory. From the general cubic I have deduced results, a con- 

 cise statement of which is subjoined. Differentiation with respect 

 to the independent variable I denote by accents, in conformity 

 with the usage of Professor De Morgan. 

 Starting with the cubic 



ax 3 + Sba^ + Scx + d=0, (1) 



1 write a = ac -b*, 

 h = od—bc, 



A= -«b 3 + 36 . 2a . b 2 -3c . 2 2 a 2 . b + d. 2 3 a 3 , 

 j3 = ba'-ab' i 



ry=Ca l — cb' } 



8=l{da'-ad') > 



B=2(m(2bdL — ah), 



C = 12&V-8tf6ab + « 2 b 2 , 



D = (12£c-«)a 2 - 66 2 ab + cZ>b 2 , 



E = B5 + C 7 + D/3-^ (B 7 +C/3) + 9 ^ 8gg Bft 



F = CS + D 7 - — (B 7 + C/3)-f- 9Z>C T^ B& 



a cr 



G=DS- i (B 7 + C/3) + ^B& 

 • H = AE' - - {6bW - 3aEF), 



1 = AF'- - (6cE 2 -2«EG-flF 2 ), 



J=AG'--(2^E 2 -«FG); 



and, eliminating a? 2 between 



A#'=E,z 2 + Fz>+G, 

 and kV + KA!x , _ H ^ 2 + 1(£ + j^ 



I find K^' + W + M*=N, (2) 



wherein 



K=A 2 E, 



L=AEA'-AH, 

 M = FH-EI, 

 N = EJ-GH. 



Manchester,' vol. ii. pp. 181, 199, and 237. Mr. Harley's results are cha- 

 racterized by great simplicity and beauty, and by a striking peculiarity of 

 their Boolian form. 



