138 Proceedings of Royal Society of Edinburgh. 
?n 1 [],l] + m 2 [l,2]+ +m n [l,w] = 0 
[sess. 
m-fnX\ + m 2 [w, 2] 4 4 mfn,ri] = 0. 
Uow, since the first column in A is composed of units, we have 
r=n 
y [r,s] = 0 for all values of s greater than unity. Therefore 
r = 1 
^£[r, l]-0; that isA =0. 
r = 1 
But A = 0 is the condition that 
yd 1 4 oq 4 d* 4 .... + = 0 , 
7i 2 4 Gq 4 d* C 3 /X 2 + . . . . 4 tn T 2 ~ 0 ) 
+ aq 4 b 2 \ n + c%g n 4 . . . . + t n r n = 0 ; 
which means that nf, . . . , n n 2 , must have the common 
value n 2 . 
12. As a special example, leading to easily calculable numerical 
values which might serve for the specification of mechanical models, 
we may consider the scheme — 
1 X fX 
T 1 2 
1 2 1 
1 0 3 
Using these values in equations (3), together with the conditions 
for observance of the third law of motion, we get seven of the 
nine constants given in terms of the remaining two and the mass- 
ratios. The results are— 
6 t-i = (Xq 4 
c,m, - m. 
^3 j a 2 ~ a z d - 
m 9 - uio , 
— -d Q 
m, m Q 
b\ — ~ 3^3 j ^2 ~ ~ 3~dz ) c '\= ~ 3~^3 > 
m, rrio 7 
Co— ~ 3 — do , d., = —do. 
Since A is zero, the As in § 6 must be zero ; so that, in equations 
