PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



261 



see the reason of this analogy. We shall, in Example 7, exhibit a complete and 

 general solution of aU equations of this form. 



Ex.6. 

 Let 



y — mx^ — =j^ =0 



then 



x^ n^ x^ « «■ 



+ ^ + -77 + &C. 

 X^ Xi 



Aa7- + Ba; + C+ ^ + &c. =m\/^l(^x'K.\S^^\ 



■»* V/A / 



Vi 



- A /i _ A, /3 



1 wV-l/2 



wV-l/3i 



n:^ — » A „ — 



A, /5^ 



8\/-l /6 



^&c. 



B =_1^ /I B - ^^ /6j 



&c. 



C, = ^=4c.= -gwl,C,= ^^/I^&c. 



This gives us six separate series. 



r. A«:2+-^ + ^*+&c.= A(x' ^-2 



Let 



V-1 /8 



+ 



1.2.6.7 



dx^ ^ i.i.im^x^ 



* . f . f »»-' ^3 -r i . S . I- . V • ¥ • ¥'»' ** 

 1.2 



+ &C.) 



+ &c. 



then >/, = s--r 



+ 



1.2 



3.4 i ■ i- . Jwi^iT "^ i . f . 4 . V . ¥ . Vw^ir'' 



1.2 



&c. 



c?^« 3.4 m'xi i.i.im'x'^' 



105 



■ir- + 



1 rf2 



^1 



or 



or 



32 OT^^V <^a!2 



rf^ 3rf^__3_rf^ _3_ _105 1 d'y^ 



dx' 2x dx^ 4: x^ dx "^ 8 x^^^~ 32 '^ m^a^'d^ 



d\y (3 1 \dr-j,, 3 dy 3 _105 



dx^ '^ \2x nfix^) dx^ 4«'' dx "^ 8^^i- "32'"' 



2°. ^ + ^ + &c.,giyes-^ a(1 -- 



"* X-- 'o mV-1 \a!* 3. 



— + 



4.5»?2aV 3.4.5.8.9.10»?2aV 



T-TT+fcc 



■) 



Let 



VOL. XVI. PART III. 



d x^ xh 3 . 4 . 5 OT^ 



hry:.+&'^- 



3 [J 



