PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 995 
—(e™ cos a—e” 4/ —1 sina—1)" (cos az—V/ —1 sin az)} 
Let e™ cos a—1=P cos 0, e” sin a=P sin 0; 
a 
then P2—e?™ Je" cosa+1, and tan 0=—° 9? 
ée” cos a—i 
me x ee, ——— = 
o. A®e™? sinaw=——-— P” {(cosm2AT+04+V—1sinn2A7 +0) (cosax+V/—1 sin a2) 
27-1 
—(cos2N 7+ 0—V/7 —1 sin 2 7 + 0) (cosaz—V/—1 sina 2)} 
e”™ 

= {W/ =Te0s n (N—D) w+8in n (WD) Ttsin(ax+nO0+nX+N T) 
=e"* P® (cosrnT—V—1 sinrnT) sin (av+n0+nAT) 
rand A being any integers. 
Similar expressions may be obtained for the wth differences of cos ax and 
of e”* cos ax. 
25. We shall now proceed to the demonstration of certain theorems analogous 
to those in the ordinary calculus of differences. 
it 1 nb ai n(n—1). 6 





Pp. l. = — &e. 
Pro Vn +n Vy Vx Vx+1 Vy Vx 41 Vz42 
where ¥,=a+ 62 OF Ave=d. 
5 1 
By formula (6); putting — for «, 
x 
1 1 1 n—1 1 
ce ba Cee ) 7 + &e. 
Opies Oe Vy ee? 0; 
N a 1 A Uz b Ae 1 IL Bs fs 
ow = + ae ie eS 
e Vy Vx41 Vy Up41 Vy Vy Vez4+1 Vx42 rae 
1 1 b —1)2 
ERE eh OP a. 
Vpin Vx Vy Vy 41 Vy Vp41 O42 
Prop. 2. A similar result may be obtained from formula (8). 
1 b Pet Puh SIR 2 Noe 



For A =— eA = &e. 
Vz—1 Vg Vy J Vy, 2 Vy Vz—Y Vz 2 
1 it nb n (n+1). 6? 
oe ee pera 
Uxz+n Vx Vz Vz_] Vy Vz_} Vz_—2 
n a n n—1 
PRop. 3. A” Uy, 0, =0, A” u,+W hv, A” u,, + &e. 
For {(1+ 4) (1+ 4’)—1}, v, being an operation on “, % may be repeated accord- 
ing to any law, consequently 
A" uw, 0,={(1+ A) (1+4’)—1}” w, v,: and every step in the demonstration is the 
same as when » is a whole number. 
