IN THE DIFFERENTIAL CALCULUS. 49 
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a result which is true, whether 7 be integral or fractional. 
The value of from Equation (4.) is easily found, and is given by Mr BooLe 
in the form 
w=(1+ w)"**¥ (p)+(1—p)"** x(¢) 
whence that of w is found. 
Let us take, as our next example, the equation which has been discussed by 
M. Potsson in the Journal del’ Ecole Polytechnique, cah. 17, p. 614; and by Profes- 
sor Boote in the Philosophical Transactions for 1844, p. 254. 
pti 
x 
Ex. 4. (1-2) S44 { —(4n—p+l1) 2 \T-2n@ n—p)u=0.... (1) 
The symbolical form of this equation is 
yD t2n—2) (D+2—2—p) RY. 


D(D+p) 1) enna) 
Let u=f(-g)» J dee 6 (GBS adaeint 
f (-3)e a! a ee ae : +1) e?4v=0.. (4) 
This last equation may be reduced to an integrable form in various ways : 
1. By making /f(- y +1) = 

D+p D+2n-—3 ( >) 



D—-1° D+2n—2—p aD) 
i D p 3 
fs pee a ee, — — 
, ea PA 2 ay.) aioe ee 
ia 2 ee: LS (2D So j/ sb 
aw Wh ks 2 
eee 1 [EDs at 
bird ye IT 2) 12 eo an—p-De] 2 2 2 p(2n—p—2) 6 
(eee ED 
) 2 | 2 
eavatal mea 
a Pp -P ad 2 pat n+2 ld 2 2n—p—2 
or “= —2) x C a) x G =) iz 2 
and equation (4.) becomes 
_ (D+2n—2) D+2n—8) 
D(D-1) 
which is a known form, and thus the given equation is completely solved. 
It is evident that our solution reduces the operation to that of ordinary dif- 
ferentiation or integration, when p is an odd integer, positive or negative. By 
varying the process, however, we can obtain other forms of the solution of this 
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