104 Mr murphy, ON THE RESOLUTION OF 



SECOND CLASS OF EQUATIONS. 



Given (p {ii.„ «.+i, u.,+2---'---Us+„} =0. 



Put //, =^, + By' + c,{Byy +c,(BYy kc. 



u,,, = A, + y.BY + C,y-.{By'f +C,Y{Byy &C. 

 U,,, = A,+Y.By' + C,y'(By'r +C,Y(Byy &C. 

 &c. = &c. 



and = (p{Ai, Ai, A3 ^„+i}, 



and ultimately make ^,=^. = ^3= ^»+i. 



Substituting in the proposed equation we have 



<!> + r, . ^7' + r. . (By'^Y + T, . (By'Y + &C. = 0, 

 d^ d<i> „ rf* „ rf'I' 



1 frfM) „ rf'4' , rf-O „„ </'<!' 1 



&c. = &c. 

 Hence O = 0, r, = 0, r, = 0, &c. 



the equation 4> = (putting A^ = A->= A,,^,) determines A,, 



\\ = will give V different values of 7, any of which may be used, 



Tj = will determine c , , 



Fa = c,, 



&c. 

 and B will remain an arbitrary constant. 



Now in linear equations the sum of the particular solutions gives 

 the complete integral, but this is not generally the case in other in- 



