8 Mr. C. J. Hargreave on the Integration 



in which, beside the law of combination, 



the further condition was imposed, 



9r,„7r„ = 7r„ 7r„, — a {m — n) p. 

 This solution, which will appear in the Cambridge and Dub- 

 lin Mathematical Journal for January 184-7, involves the so- 

 lution of a class of differential equations of which that of La- 

 place's functions is only a very particular case. Such methods, 

 limited in their individual application, and apparently indefi- 

 nite in their number, seem however to be chiefly valuable as 

 exercises in symbolical algebra. Linear differential equa- 

 tions, and linear equations in finite differences, may, as I have 

 elsewhere shown, be reduced to the general form 



?i + <p,(7r)p?< + (p2 (w) ^^M... = U, 

 in which tt and p satisfy the relationsy(7r) p 7i = pJ'{7r-\- l)u and 

 J'{ir)p'" =f(}n)p"' ; and additional experience confirms my be- 

 lief, that the methods which are founded on the employment 

 of this form are sufficient for every case. Mr. Bronwin's 

 equations, when thus treated, are at once seen to be inte- 

 grable. 



It is but justice to add, that Mr. Bronwin's method displays 

 considerable ingenuity. 



I remain, Gentlemen, 



Your obedient Servant, 

 Lincoln, Dec. 4, 1846. George Boole. 



IV. Observations on the Rev. B. Bronwin's Paper on the In- 

 tegration and Transformation of certain Differential Equa- 

 tions. By Chari.es James Hargreave, F.R.S. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



I TAKE the liberty of drawing your attention to an error 

 which appears to exist in the paper contained in your 

 December Number on the Integration and Transformation of 

 certain Differential Equations. 



I confine my observations in the first place to the first of 

 the equations discussed by the author ; but it will be seen 

 that the same or similar remarks are applicable to the other 

 equations. 



The solution of the equation 



■(g+^^O + ^^^o CO 



