278 Integration of Differential Equations. 



d^z dz 2u 



We will now reduce the equation u -j-^-{--T--2a'uz-Tj-, {m), 



d'^z dz 

 to the form of (2); (m) is easily changed to J~i+~7~ — 2a' 



/ , 1 \ 1 d^y dy 



V^-^'J =0' ^^^ '^ ^" P"^ 2/=^+^ we have ^.+,7^- 

 2a'y=0, {n). If we put 2g-2 = or ^=1, a2=l, h-=2a', 

 c—lo'c:2qp — q-{-l—2p = \, we get ^=^, and (2) will become 

 the same as(w). Hence if in (e) we put g'^l, a^=\^ h—V —2a', 



we have y=K' f dx{l-x'')-^cos.ux'^ - 2a'+By dx{\-x^)-^ 



log.[M(l — a;^)]cos.2ia;v^— 2a/, (0), for the complete value of y, 

 which is the integral of in). If we use e for the hyperbolic base, 



- xuwia' , uxw 2a' 



/ e +e 



we get by known formulae cos.wa;v — 2a'= q j 



and if yre put a:=:cos.qD we get dx{l—x^)'^= — d<p, l—x^=^ 



-uv 2a'.cos.(p . M ■v 2a'.cos.(p 



e ^ + e 



sin. 2 9, cos.iia;V'~2a'-— ^ . by substituting 



^, , • / \ 1 /* 7 / -Mv 2a'.cos.(p , wv2a'.cos.m\ 



these values m(o) we have 2/= — / agcle -\-e 



/A'' B' \ 



(-o-4-"o" log.Msin.^^qo), the integral being taken from cos-^^O to 



A' B' 

 cos.9=l ; or if we denote -^ and -g, by A and B, we have by 



interchanging the limits of the integral and changing its sign, 



(which makes no alteration in its value,) 2/=/ dcpie'^ 



4-e" '^■cos<p| ^ fA+Blog.wsin.^qDJ, the integral being taken from 



cos. qo = 1 to cos. <p = 0, or if we put ^ = 3.14159, &c. (= the 

 semi-circumference of a circle whose radius =1,) we get y~ 



/TT I -u'>/2a'.coa.(p , u'/2a'.cos.q)\ /..■oi • n \ n j^ 



3(ZqD e +e J . (A+Blog.wsm.2(jD , and 5 



denoting the limits of cp; or (since cos.go becomes negative in the 

 second quadrant) we shall obtain the same value of y by putting 



y=J d^>e '"■'^°^*''[A+Blog.wsin.2 9J, (w'), the integral being 



taken from 9 = 0, to 9='t, {m!) will be found on trial to satisfy 

 {n\ and since it involves the two independent arbitrary constants, 

 y is the complete integral of (w) as required. 



