274 Integration of Differential Equations. 



(1) is taken from Vol. Ill, p. 537, second edition, of the Traite 

 du Calcul Differentiel et du Calcul Integral, par S. F. Lacroix ; 

 and (2) is deduced from (1) by changing the sign of the last 

 term, or which comes to the same thing, by changing h^ in (1) 

 into —6% that is, using hV — 1 for h. We shall put 2pq-q-\'l 

 =c, (a), and shall use the characteristic,/^, when prefixed to any 

 differential expression, to signify that its integral is to be taken 

 with reference to the variable, from its value ti, to the value m; 

 or n and ni are two values of the variable, and the integral is sup- 

 posed to be taken from the first limit n, to the second m. 



/a p- 1 



dx{a^ —x^) cos. bu^x, is a par- 

 ticular value of y, which satisfies (1) ; in which u^ is to be re- 

 garded as constant in taking the integral, x and its functions be- 

 ing the only variables, and the integral is supposed to be taken 

 from x=0 to x = a, which are the first and second limits of x. 



We have also found by integrating (1) (by the method of se- 

 sies) that it is satisfied by the particular value of y, which is de- 



notea by 2/=w'~y dv{a--^v'^) cos.bu^v, when p is positive, 



and such that 1 —p is positive, and not an indefinitely small quan- 

 tity, and it is to be noted that v and its functions are considered 

 as the only variables in taking the integral, u^ being regarded as 

 constant. Hence if we use A and B to denote two arbitrary con- 

 stants, we shall have (by the well known theory of integrals) for 



/a p- 1 



dx{a'^ —x^) 



1 - c /»a -p 



cos.bu'^x-{-'Bu j dv(a^—V") cos.bu^v, (&), in which jd must 



not be considered as an indefinitely small quantity, and 1-^ is 

 positive and finite ; since the limits of x and v are the same in 

 (6), we may change v into x, and then the value of y may be 



/a p-1 I \-c 



dx{a^ — x^) cos. bu'^x A + B w 



{a^ -x"^) j, (c). If we put 1 — c=e, 1 — 2p=/, orc=l— e, 



l-c l-2p / 



2p = l— /, {d), we get u [a^ —x^ ) =u%a^ -x") , and if we 



substitute the values of c and 2p from (d) in (a), we get by a 



e f I IV 



slight reduction /=-, .'. u^(a''—x-) = \u{a^~x^)^] , hence, 



when e is very small, using hyperbolic logarithms and rejecting 



