55 



■^The expression (3) cannot be considered as including the 



case of the Hniit, for the general expression is 



r, , /C"'^ sin Rx \ C5) 



c^ e»-^cos/3a: -r c A 1 J '^ ^ 



now wliatever value we assign to /3 excepting /S — o, as c, is 

 arbitrary -^ is arbitrary, and therefore may be expressed by c.. 

 But in order that the expression may be general, it must be retain- 

 ed of the form (5). Then if we make |S=o it becomes 



e^^Fc, + c, ii[L!?f"| or c^ 6*^+ cc^-r because the limit of 



Sin ^jT 



I shall not give here the investigation of the form (5) but only 

 mention that I deduced it by a method of integration similar to 

 that which I applied to the differential equation of the Lunar 

 orbit in a preceding paper. By which method I obtain the gene- 

 ral integration of Equat. (1) Avithout the intervention of impos- 

 sible expressions. 



Again, if 1+^^'+ J8x'-I- - - Pa." contain two equal factors 

 of the form (1— ax)- +/S'a;- Euler first and lastly Lacroix* 

 have made the corresponding part of the integral 



e« j (c. + c, ,r) cos /3 X + (Cj+ Cj^) sin /3.r (6) 



now when jS = o 



this becomes e''-^ (c^-V c, .r)Vhereas the part of the integral aris- 

 ing from four equal simple factors of the form ( 1 —a. x) is 

 e'^^ (c, + Co X + Cj J" + c, 2-^) as is well known. 



Here then the same difficulty occurs, and the method of li- 

 mits might be considered as leading in this instance also to an 



» Lacroix Traite du Calcul. Diff. &c. torn. 2. p. 319, ed. 2. Tlie first edition has not the 

 case of the equal quadratic factors. 



