IN THE DIFFERENTIAL CALCULUS. 43 





/=2 +7 
/ 1 ree men te tt 
= — x Loe foes Eee SE 
ee {3 2 r+2—rpe [r+2 
i wee 
rol a ee (r +2) (27+ 2) Re } 
r+2 (@+2—rp)Qr+2—rp) 
3 j—e+ : 3 
ORT: re tY\ i eal (eee axe" r4+2 ; 
sre ke hee ix {; 2 jr+2 r+2—rp +&e. } 
[oe 
SONY Soa () 
The same process will give the same result for all the terms of this expan- 
sion except the last. But as mae is a factor of that term, it is evidently 0. 

: Ae eee! k ; 
Consequently, the equation & = (« iC a2) =a'x'“y gives the complete value 
fe an 
of y=d' e**, it being observed that one of the constants of integration is 0. The 
other constants must be determined by means of equation (A). 
10. To find a cos az. 

2 4 
Since cos ar=1—5* +e we. } 
if —a’=6, we shall have 
ge ds\ jee 
6 7 
—_— aay getter) Celie DAREN BOE 
(5 x) cos ax=(—7) {a a, gear 
ieee 
: =e [=6+7rp 
ei 

fag Bl or 6— 
a : [ante Me a Ae Ee 7 be a 
(Srimgst ese e ay 
ie ‘ | Te 
It remains to express this series in the form of a differential equation when 
r is a whole number. 




/-2strp 
Let s be any whole number less than r+1, S=(—7)“s° ___”____; then the 
| 2s 
| aa 
series may be written 
reigznenee or (r+2s)(27r+2s) Denoen. 
y=8{ j2s + rte (4+2s—rp)(Qrt+2s—rp) i be. | 
Got ( ae ~) (fash pete a” as 
oo (2"" 2”) =s 2 4) oe 
dx dz 2s j2s—r /2s+r—1 r+2s—rpu 
VOL. XX. PART I. M 
