362 
PROFESSOR A. R. FORSYTH ON 
aljy) = cr„ 
aZ„,(w+'y) = S„ 
and regard cr m and as being of the same order as s m . Then accurately to the seventh 
order 
al m (u)al m (v)al m (u+v) 
_ O 
1 —-g “'Z m U/, w(^,~ + cr, ' + S,-)—-y- X" 1 & r , + CT, 1 -f- S, 4 ) 
5-=l 
r=p 
5=1 
+ 5 a,._ OT a <: m (s / 2 s t ~ -j-ay <x f +S,- 2 S t 2 ) + 2 ™ a ri ^^(.yoy+ovSpd-S,- 2 .^ 2 ) 
?•,£=! r=l t= 1 
(32) 
where the summation in the last term in (32) is exactly as in the last term in (30). 
To express this in terms of u and v we must substitute the value of s in terms of us as 
given by (29) and for a and S respectively corresponding values of v and u-\-v. Let 
these values be inserted, both sides multiplied by and the summation taken for 
the values m— 1 to m=p and compare this expression, which is 
5il = p l 
t ppr 1 — al m (u)al m (v)al m (u+ v) 
m=l \flm) 
(B). 
with the value of U + Y+W. 
Firstly, they agree in the third order of quantities; for 
uj + vj + wj 
- ^mVm {u m | t in) 
since 
u m 4~ v m -f- w m — 0. 
Secondly, consider in each the terms of the order five. That in U-j-V+W which 
has l'' 1 : K m for its coefficient is 
I («,«) 
-ack + v ») h — u >* — v >*~\ 
=fumV M (u m 3 + 2u„?v m +'2u m v m 2 + v m d ) 
— 3" W/itV/n(u t// —b V/,i) ( U„: ~b V nl d - Mintin') 
3 UmVm{u M + V M ) { U -J + V m 2 + (u„, + V m )' 2 } 
while in (B) it is 
and these are obviously equal. 
The term in U + Y + W which has 
L 1 
1 Ir 
r 3 
P'(0 ?'(">■) co r -aJ Le - 3 ¥'(a m f r ’ m ° V 3 P'( a r f m > 
