and on Co -resolvents. 181 



Again, when 2^=?^ — 2, we have, subtracting as before, the 

 condition 



a (X—l) + (3 (/t— 1) + 7 (''—I) -^ ... - 2 - (.-) 



of which the only available solutions are of the forms 

 \ = 3, ju. = I, V = I, p = 1, . . a = 1, /3 = 1, . . . 

 and 



\ = 2, /a = 2, i^ = I, p = I, , . a = 1, fi = I, . . , 



Hence 



and we see that the conditions for the annihilation of the 

 second and third terms of an equation of any order will not 

 essentially differ from those for equations of the third order 

 already discussed. 



The forms of the conditions also show that the simul- 

 taneous destruction of the second and rth terms of a linear 

 differential equation of any order may be made to depend 

 upon the solution of equations of the first and 7'th order : 

 and thus far the analogy between algebra and the calculus 

 holds — to a cei-tain extent at least, I have not ascertained 

 whether the resulting equation in the case of r being greater 

 than 2 can be made linear. 



12. The sinister of a linear (or simple) algebraical equa- 

 tion whose dexter is zero may be reduced to a single term 

 by an easy transformation, and the sinister of a linear 

 differential equation of the first order whose dexter is zero 

 possesses the analogous property that, being multiplied into 

 an appropriate factor, it may be reduced to the form of a 

 perfect differential coefficient. We will now inquii'e whether 

 the analogy between algebra and the differential calculus 

 holds in the case of the next degree and order. And in so 

 doing it will no longer be necessary for me to adhere to the 

 abridged notation so conducive to the perspicuity of fore- 

 going results and to the manageability of the formulae in- 

 volved in them. I proceed to attempt the annihilation of 

 the last two terms of a linear differential equation of the 

 second order, and for a reason given in Article 2 I shall 

 start from the linear differential equation of the second 

 order 



g + >-. = o - - - (1) 



